Maurice Pouzet

WELL-QUASI-ORDERING AND EMBEDDABILITY OF RELATIONAL STRUCTURES

Keywords: Mathematics Subject Classification. 03 C13, 03 C52, 05 A16, 05 C30, 20 B27 Relational structures, ages, counting functions, oligomorphic groups, age algebra, Ramsey theorem, well-quasi-ordering, cellular graphs, tournaments

At the end of the forties, Fraïssé, following Cantor, Hausdorff and Sierpinski, highlighted the role of the embeddability quasi-order in the theory of relations. Since then, many results illustrating this role have been obtained (a large account was included in Fraïssé's book Theory of Relations). In this paper, I present a selection of results centered on the notion of well-quasi-order (wqo). I mention several problems on wqo and hereditary classes of relational structures; some of these problems go back to the seventies.

INTRODUCTION

This paper is about relational structures, like posets and graphs, and the embeddability quasi-order. We will present a selection of notions and results collected on the road opened by Roland Fraïssé. The basic notions are those of embeddability, hereditary classes, ages and bounds, the results being mostly about hereditary classes, particularly on those which are well-quasi-ordered (wqo for short) or even better-quasiordered (bqo). We will mention problems, some going back to the seventies, on wqo and hereditary classes of relational structures. The concepts presented here have an infinistic nature, but the emphasis is put on the comparison of finite structures by embeddability or by set inclusion, rather than the comparison of infinite structures. Few comparability results on infinite substructures are summarized in Subsection 2.1 and in Section 8. An important area of the theory of relations is the reconstruction of relational structures (in particular, of finite structures) as introduced by Roland Fraïssé and developed by Gérard Lopez and his collaborators. The latter is not covered in this paper and we refer the interested reader to [START_REF] Fraïssé | Theory of relations[END_REF][START_REF] Fraïssé | La reconstruction d'une relation dans l'hypothèse forte: isomorphie des restrictions à chaque partie stricte de la base. With an appendix[END_REF][START_REF] Dammak | Hypomorphy of graphs up to complementation[END_REF].

A relational structure R is embeddable in a relational structure R ′ if R is isomorphic to an induced substructure of R ′ . This defines a quasi-order on the class of relational structures. At the end of the forties, Fraïssé, following the work of Cantor, Hausdorff and Sierpinski, highlighted the role of this quasi-order in the theory of relations. For example, he conjectured that the class of countable chains is well-quasi-ordered (wqo) under embeddability, a conjecture positively solved by Laver [START_REF] Laver | On Fraïssé's order type conjecture[END_REF] in the early seventies with the use of the theory of better-quasi-ordering (bqo), a far reaching strengthening of the notion of wqo, invented by Nash-Williams [START_REF] St | On well-quasi-ordering infinite trees[END_REF]. Fraïssé also noted that basic notions about ordered sets (posets) like initial segments, ideals, chains and antichains have a direct counterpart in terms of relational structures. For example, a class C of structures is hereditary if it contains every structure which can be embedded into some member of C. Clearly, hereditary classes are initial segments of the class of relational structures quasi-ordered by embeddability. If R is a relational structure, the age of R is the set Age(R) of finite restrictions of R considered up to isomorphy. This is an ideal of the poset made of finite structures considered up to isomorphy and ordered by embeddability. As shown by Fraïssé in the fifties, every countable ideal has this form. Several results concerning hereditary classes of relational structures and based on their order properties have been then obtained. Recent years have seen a renewed interest for the study of hereditary classes made of finite structures and notably for their profile. The profile of a hereditary class C of finite structures considered up to isomorphy is the function φ C which counts for every nonnegative integer n the number of members of C on n elements. General counting results as well as exact results for graphs, tournaments, ordered graphs and permutations have been obtained, with a particular emphasis on jumps in the growth of the profile.

The content of this paper following this introduction is divided into nine sections. In the first one, I present the notions of order and topology on which our approach is based. Due to the survey nature of this paper, I slowly introduce the basic notions of the theory of relations, like relational structure, homomorphism, isomorphism, giving several examples, like graphs, posets, hypergraphs, words, permutations and bichains, metric spaces, codings of permutation groups. In the following section, I introduce the quasi-order of embeddability and compare isomorphy to equimorphy. Next, I introduce hereditary classes, ages, bounds, and the link with model theory via universal classes. I illustrate these notions with several examples. Then, in Section 3, I consider a classification of hereditary classes based on the notion of well-foundedness. It leads to the notions of wqo and better-quasi-ordering (bqo). I present some of the major results and problems about wqo hereditary classes and ages. Notably, I discuss the use of order and topological means (for example, height, ordinal length, Cantor-Bendixson and Vietoris ranks). In Section 4, I discuss the preservation of well-quasi-order by labelling. There are several notions, some could be equivalent. When they hold, this has some strong consequences, notably on the finiteness of bounds of hereditary classes. In Section 5, I present a classification of hereditary classes and ages of small height. The main result is used in Section 3 and appears also in Section 6. In Section 6, I present the profile of a class of finite relational structures. I discuss the effect of the wqo character of the class under consideration on its profile (for example, a description of hereditary classes of small growth). In Section 7, I look at the number of wqo hereditary classes. With the use of uniformly recurrent sequences I report on the fact [START_REF] Oudrar | Minimal prime ages, words and permutation graphs[END_REF] that the number of wqo hereditary classes of permutation graphs has the cardinality of the continuum. Looking at the links with the notion of primality, I present the notions of minimal prime structures and describe minimal prime hereditary classes of graphs, such a description is based on a result of Chudnovski et al [START_REF] Chudnovsky | Unavoidable induced subgraphs in large graphs with no homogeneous sets[END_REF]. In Section 8, I discuss how Laver's theorem on scattered chains could be extended to other classes of structures. A brief section ends the paper. Only some proof are given, preferably the simpler ones in order to give a feeling of the subject. Some proofs given did not yet appear in a refereed journal.

This work is the third, after [START_REF] Pouzet | Applications of well-quasi-ordering and better quasi-ordering[END_REF][START_REF] Pouzet | The profile of relations[END_REF], of a survey nature that I wrote concerning wqo classes of relational structures. It is dedicated to the memory of Roland Fraïssé. Without him and the contribution of many other people, colleagues as well as students, this text would not exist. They are cited in the text and in the references. I thank them very much.

This paper is an outgrowth of the presentations made in the Mini-symposium 38 "Well-quasi-ordering: from theory and applications", Hamburg, September 21-25, 2015 1 and in the seminar "Well-quasi-Orders in Computer Science", Dagsthul, January 17-22, 2016 2 . I would like to thank Professor Monika Seisenberger and the organisers for their warm hospitality. The conference "Algebras, Graphs and Ordered Sets" (ALGOS 2020), Nancy, August 26 -28, 2020, gave me yet another opportunity to present this subject. I warmly thank Miguel Couceiro, the general chair, and the organizers for their kind attention. I also wish to express my gratitude to both editors of this volume, Dwight Duffus and Miguel Couceiro, for their tireless work on this paper.

THE FRAMEWORK

Several notions about quasi-orders, or rather orders, like initial segments, ideals and antichains, yield natural classes of relational structures. In fact, initial segments and ideals of the collection Ω µ of finite relational structures of a given signature µ, quasi-ordered by embeddability, are the basic objects on which relies any combinatorial or model theoretic study of finite relational structures. Besides, many concrete classes of structures fall into these categories. Also, there is an interplay between properties of subclasses of Ωµ , the class of all structures of signature µ, and their traces on Ω µ . In a lesser extent, ordered topological spaces arise in this context. Some of the basic notions needed are recalled below. The notion of well-quasiorder, quite central in this study, will be presented in Section 3.

1.1. Quasi-ordered sets, ordered sets, chains and antichains, wellfoundedness, initial segments, ideals, final segments, filters. A reflexive and transitive relation ρ on a set V is a quasi-order, the pair P ∶= (V, ρ) is a quasi-ordered set (quoset); if in addition ρ is antisymmetric, P is a (partially) ordered set (poset). If P is a poset or a quoset, we denote by x ≤ y the fact that (x, y) ∈ ρ. The dual of P ∶= (V, ρ) is the quoset P dual ∶= (V, ρ -1 ) where ρ -1 ∶=∶ {(x, y) ∶ (y, x) ∈ ρ}. The distinction between quasi-order and order is (in part) a matter of convenience. Indeed, if a set V is endowed with a quasi-order, denoted by ≤, the binary relation ≡, defined on V by x ≡ y if x ≤ y and y ≤ x, is an equivalence relation. The image of the quasi-order on the quotient set V / ≡ is an order and (V, ≤) is the lexicographical sum of the equivalence classes indexed by the quotient, denoted by (V, ≤)/ ≡. So the properties of the quasi-order reduces to those of the quotient. In the sequel, We will mostly work with posets.

Let P ∶= (V, ≤) be a poset. Two elements x, y ∈ V are comparable if either x ≤ y or y ≤ x. A subset A of P is a chain if any two elements x, y ∈ A are comparable. On the opposite, A is an antichain if distinct elements x, y of A are incomparable. The poset P is well-founded if every nonempty subset has a minimal element; if this is the case, the height h(x, P ) of an element x ∈ V is the ordinal defined by transfinite induction by the formula:

(1.1) h(x, P ) ∶= sup{h(y, P ) + 1 ∶ y ∈ V, y < x}.

The height of P is h(P ) ∶= sup{h(y, P ) + 1 ∶ y ∈ V }. For each ordinal α < h(P ) the α-level of P is the set P α ∶= {x ∈ V ∶ h(x, P ) = α}. The poset P is level-finite if each of its levels is finite. König's theorem asserts that: every well-founded level-finite poset contains a chain going throught all the levels. If P ∶= (V, ≤) is a poset, a subset I of V is an initial segment of P if x ∈ V and y ∈ I with x ≤ y imply x ∈ I. If A is a subset of P , the set ↓ A ∶= {x ∈ V ∶ x ≤ y for some y ∈ A} is the least initial segment containing A, we say that it is generated by A; if A = {y}, Set ↓ y instead of ↓ {y}, this is the principal initial segment of P generated by y. An initial segment I is finitely generated if I =↓ A for some finite subset A of V . Denote by down(P ) the set of principal initial segments of P . Denote by I <ω (P ) the set of finitely generated initial segments of P . The set I(P ) of initial segments of P , once ordered by set inclusion, is a complete lattice, the join being the union and the meet being the intersection. Among complete lattices, I(P ) is characterized by the fact that it is completely distributive and generated by its completely join-irreducible elements. The order on I(P ) determines the order on P since the completely join-irreducible elements of I(P ) are the principal initial segments of P . An ideal of P is any nonempty initial segment I which is up-directed (that is for every pair x, y ∈ P there is an upper bound z ∈ I, that is, satisfying x, y ≤ z). As it is well known, ideals of P coincide with the join-irreducible members of the lattice I(P ), that is the nonempty initial segments I such that I = I ′ ∪ I ′′ implies I = I ′ or I = I ′′ . Denote by Id(P ) the set of ideals of P . Note that if P is a quasi-ordered set, initial segments are simply the inverse images of initial segments of the quotient V / ≡. Final segments, resp., principal final segments, of P can be defined as initial segments, resp., principal initial segments, of V equipped with the dual order ≥. Set ↑ A ∶= {x ∈ V ∶ x ≥ y for some y ∈ A}. Denote by ↑ y the principal final segment generated by an element y ∈ V . Denote by up(P ), resp., by F <ω (P ), resp., F(P ), the set, ordered by inclusion, of principal final segments, resp., finitely generated final segments, resp., final segments of P . Filters can be defined as ideals of the dual of P . Denote by Fi(P ) the set of filters of P . Every antichain A yields two initial segments, namely P ∖ ↑ A and ↓ A; and A can be recovered by looking respectively at the minimal elements of ↑ A and at the maximal elements of ↓ A. If P is well-founded or dually well-founded then every initial segment is determined by an antichain.

The study of initial segments and ideals has many similarities with the study of varieties in ring theory, hereditary classes playing the role of ideals in ring theory and ideals the role of irreducible ideals. In particular chain conditions, and more precisely well-quasi-order, play an important role. These aspects were noted by Malcev, as well as Robinson and Fraïssé [START_REF] Mal'cev | The metamathematics of algebraic systems[END_REF] [START_REF] Robinson | On the metamathematics of algebra[END_REF], [START_REF] Fraïssé | Theory of relations[END_REF]. More recently, Cameron, introducing the algebra of an age [START_REF] Cameron | The algebra of an age, In Model theory of groups and automorphism groups[END_REF], gave extra support to this approach. In the sequel, We will illustrate the role of initial segments and ideals, introducing other order theoretic notions when needed. 1.2. Topology and order; scattered ordered sets and scattered topological spaces. A topological space is called scattered if every nonempty subset has at least an isolated point. The Cantor-Bendixson procedure begins with a topological space X and associates to each ordinal number α a subspace X (α) , the α thderivative of X, defined as follows: X (0) ∶= X, X (α) ∶= {x ∈ X (β) ∶ x is not isolated in X (β) } if α is a successor ordinal, α = β + 1, and X (α) ∶= ⋂{X (β) ∶ β < α} otherwise. An element x ∈ X has rank α, denoted by rank(x, X) ∶= α if:

(1.2)

x ∈ X (α) ∖ X (α+1) .

Since the X (α) 's are subsets of X, there is an ordinal β such that X (α) = X (β) for every α ≥ β. The least ordinal β for which this property holds is the Cantor-Bendixson rank of X, denoted by rank(X). Then, the space X is scattered if and only if X (rank(X)) = ∅. Also, if a scattered space is compact, its rank is a successor ordinal; denote by rank -(X) the predecessor of that ordinal, that is the largest ordinal β such that the derivative X (β) is nonempty. Let us note that a compact space with a countable base is scattered if and only if it is countable. Identifying the powerset ℘(V ) of a set V with 2 V , we may view it as a topological space.

A basis of open sets consists of subsets of the form

[F, V ∖ H) ∶= {X ∈ ℘(V ) ∶ F ⊆ X ⊂ V ∖ H},
where F, H are finite subsets of V . Recall that ℘(V ) with this topology is a compact totally disconnected space, in brief a Stone space.

The topological notion of scatteredness is related to the notion of scatteredness for posets. Recall that a poset is order scattered if it does not embed a copy of the chain of the rational numbers. Duality between Boolean algebras and Stone spaces provides examples of this relationship, a basic result due to [START_REF] Day | Superatomic Boolean algebras[END_REF] [START_REF] Day | Superatomic Boolean algebras[END_REF] asserting that a Boolean algebra, viewed as a poset, is order scattered if and only if its Stone space is topologically scattered. Duality, as extended by Priestley [START_REF] Priestley | Representation of distributive lattices by means of ordered stone spaces[END_REF][START_REF] Priestley | Ordered topological spaces and the representation of distributive lattices[END_REF][START_REF] Davey | Introduction to lattices and order[END_REF], involves sets equipped with a topology and an order. A Priestley space can be viewed as a collection F of subsets of a set V , ordered by set inclusion, the topology on F being induced by the product topology on the powerset ℘(V ), with the additional requirement that F is closed. The fundamental duality result is that if T is a bounded distributive lattice, its spectrum, Spec(T ), the collection of prime filter of T , once equipped with the inclusion order and the topology induced by the product topology is a Priestley space. In turn, the collection of clopen subsets of Spec(T ) is order isomorphic to T . Recall that a bounded distributive lattice is order scattered if and only if its spectrum is topologically scattered.

A basic example of a Priestley space is the collection I(P ) of initial segments of a poset P equipped with the topology inherited from the product topology on the power set and set inclusion order. We recall that if P is a poset, then I(P ) and Id(I <ω (P )) are isomorphic as Priestley spaces, a basic fact about algebraic lattices, see [START_REF] Grätzer | General lattice theory, With appendices[END_REF]. In particular, if κ is a cardinal and P ∶= [κ] <ω is the set, ordered by inclusion, consisting of finite subsets of κ then Id(P ) is isomorphic to ℘(κ), the power set of κ ordered by set inclusion.

We also recall the following result (see Theorem 1.1, p.211 and Proposition 2.2, p. 215 of [START_REF] Chakir | Infinite independent sets in distributive lattices[END_REF]).

Theorem 1.1. A poset P contains a subset isomorphic to [κ] <ω if and only if Id(P ) contains a subset isomorphic to ℘(κ).

Note also that I(P ) is topologically scattered if and only if it is order scattered (see [START_REF] Fraïssé | Theory of relations[END_REF], 6.7.4, p. 187 ; see also [START_REF] Mislove | When are order scattered and topologically scattered the same?, Orders: description and roles (l'Arbresle[END_REF] for an extension to algebraic lattices). The set I(P ) is topologically closed in the powerset ℘(V ). In general, Id(P ) is not. But, the topological closures in ℘(V ) of down(P ) and Id(P ) are the same. A poset P is up-closed if every intersection of two members of up(P ) is a finite union (possibly empty) of members of up(P ). We recall below the following result (Corollary 2.7 and Proposition 2.8 of [START_REF] Bekkali | Incidence structures and Stone-Priestley duality[END_REF]).

Proposition 1.2. The following properties for a poset P ∶= (V, ≤) are equivalent:

(i) Id(P ) is topologically closed in ℘(V );

(ii) F <ω (P ) ordered by set inclusion is a bounded distributive lattice;

(iii) P ∈ F <ω (P ) and P is up-closed.

The tail algebra of a poset P ∶= (V, ≤) is the Boolean subalgebra T ailalg(P ) of the Boolean algebra (℘(V ), ∩, ∪, ¬, ∅, V ) generated by up(P ). According to J.D. Monk (Chap. 2, p.40 of [START_REF] Monk | Cardinal invariants on Boolean algebras[END_REF]), this notion is due to G. Brenner. Denote by T aillat(P ) the bounded sublattice of (℘(V ), ∪, ∩, ∅, P ), generated by up(P ). We recall Theorem 2.1 of [START_REF] Bekkali | Incidence structures and Stone-Priestley duality[END_REF].

Theorem 1.3. Let P ∶= (V, ≤) be a poset. The topological closure down(P ) of down(P ) in ℘(V ) is homeomorphic to the Stone space of T ailalg(P ). With the order of set inclusion added, down(P ) is isomorphic to the Priestley space of the lattice T aillat(P ).

As an illustration we present: Proposition 1.4. If P is a poset satisfying one of the conditions of Proposition 1.2 then T aillat(P ) = F <ω (P ) while Id(P ) (with the inclusion order and the topology) is the spectrum of T aillat(P ). In particular, if P is countable then the following conditions are equivalent: Id(P ) is countable, Id(P ) is topologically scattered, F <ω (P ) is order scattered.

Note that Id(P ) can be countable and I(P ) can be uncountable. For a simple example, let Q be the cartesian product of the chains N and 3 ∶= {0, 1, 2}. Let P be the poset obtained by deleting the comparabilities between vertices of the form (x, 1) and (y, 1). This poset is a countable lattice (not distributive) satisfying the conditions of Proposition 1.2. It contains only two infinite ideals, hence Id(P ) is countable. Since P contains an infinite antichain, I(P ) is uncountable.

1.3. Basics on relational structures.

1.3.1. The objects and their morphisms: relational structures, homomorphisms, isomorphisms. Let m be a nonnegative integer. An m-ary relation with domain, or base, V is a subset ρ of V m . The pair R ∶= (V, ρ) is a relational structure. We set V (R) ∶= V . The cardinality of R, denoted by |R|, is the cardinality of V (R). If R ∶= (V, ρ) and R ′ ∶= (V, ρ ′ ) are two m-ary relational structures, a map

f ∶ V → V ′ is a homomorphism of R into R ′ if:
(1.3) (x 1 , . . . , x m ) ∈ ρ implies (f (x 1 ), . . . , f (x m )) ∈ ρ ′ for all (x 1 , . . . , x m ) ∈ V m . An isomorphism of R onto R ′ is any bijective map f from V to V ′ such that f and its inverse f -1 are homomorphisms of R into R ′ and of R ′ into R respectively. That is, f is bijective and:

(1.4) (x 1 , . . . , x m ) ∈ ρ if and only if (f (x 1 ), . . . , f (x m )) ∈ ρ ′ for all (x 1 , . . . , x m ) ∈ V m . More generally, a relational structure is any pair R ∶= (V, (ρ i ) i∈I ) made of a set V , the domain of R (called the base in Fraïssé's book), also denoted by V (R), and a family (ρ i ) i∈I of m i -ary relations ρ i on V . The family µ ∶= (m i ) i∈I is the arity or signature of R. If I is finite then we say that the signature is finite.

If R ∶= (V, (ρ i ) i∈I ) and R ′ ∶= (V ′ , (ρ ′ i ) i∈I ) are two relational structures with the same signature µ, a map

f ∶ V → V ′ is a homomorphism of R into R ′ if f is a homorphism of (V, ρ i ) into (V ′ , ρ ′ i ) for each i ∈ I, that is, (1.5) (x 1 , . . . , x m i ) ∈ ρ i implies (f (x 1 ), . . . , f (x m i )) ∈ ρ ′ i
for all (x 1 , . . . , x m i ) ∈ V m i and i ∈ I.

An isomorphism of R onto R ′ is any bijective map f from V to V ′ such that f and its inverse f -1 are homomorphisms of R into R ′ and of R ′ into R respectively. That is, (1.6) (x 1 , . . . , x m i ) ∈ ρ i if and only if (f (x 1 ), . . . , f (x m i )) ∈ ρ ′ i for all (x 1 , . . . , x m i ) ∈ V m i and i ∈ I. Two relational structures R and R ′ are isomorphic, and we write R ≃ R ′ , if there is an isomorphism f from R onto R ′ . We also say that they have the same "isomorphism type". The isomorphism type of a relational structure is a formal object τ (R) such that a relational structure R ′ is isomorphic to R if and only if τ (R ′ ) = τ (R). In some situations, isomorphism types have a concrete representation.

In the sequel, relational structures are considered up to isomorphism, that is, no distinction is made between relational structures that are isomorphic. But behind this is a noteworthy question: How does one recognize that two structures R and R ′ are isomorphic? For countable structures, there is a test involving infinitary sentences, but not much is really known. The question for finite structures has retained a lot of attention. The problem of deciding whether two graphs (undirected, no loops) on n vertices are isomorphic or not, is not classified: it is in the class NP, but it is not known if it can be solved in polynomial time (in the number of vertices). In 2015, L. Babai announced that this problem can be solved in quasi-polynomial time, that is in time exp(O(log(n) O( 1) ) [START_REF] Babai | Graph isomorphism in quasipolynomial time[END_REF] (see also the exposition in [START_REF] Helfgott | Isomorphismes de graphes en temps quasi-polynomial[END_REF]). 1.3.2. Restriction, local isomorphism and embeddability: a way of comparing structures. To a class of structures and a type of morphism, one may associate a quasi-order on the structures of that class, saying that a structure R is smaller than a structure R ′ if there is some morphism of the given type from R to R ′ . And one may look at properties of this quasi-order as well as the influence of some well studied properties of quasi-orders, like well-quasi-ordering, on the properties of the class of structures under consideration. This has been done for the class of (irreflexive) graphs with graph homomorphisms and with graph minors. Many important results have been obtained, see the book of Hell and Nešetřil [START_REF] Hell | Graphs and homomorphisms[END_REF] for graph homomorphisms and of Robertson and Seymour [START_REF] Robertson | Graph minors. A survey[END_REF] for graph minors, notably the well-quasi-ordering of the class of finite graphs under the minor quasi-ordering, a result of first order magnitude.

This paper is devoted to classes of relational structures where the homomorphisms are embeddings. This approach, started by Fraïssé, led to several developments, some are mentioned in his book [START_REF] Fraïssé | Theory of relations[END_REF]. This approach is based on the following definitions.

The restriction of an m-ary relation ρ to a subset A of its domain V is the relation ρ ↾A ∶= ρ ∩ A m . The substructure induced by a relational structure R ∶= (V, (ρ i ) i∈I ) on a subset A of V , simply called the

restriction of R to A, is the relational structure R ↾ A ∶= (A, (ρ i ↾ A ) i∈I ). A local isomorphism of R is any isomorphism between two restrictions of R. A embedding from R to R ′ is any isomorphism f of R onto the restriction of R ′ to the image of V . A relational structure R is embeddable into a relational structure R ′ ,
with the same signature µ, and we set R ≤ R ′ , if R is isomorphic to a restriction of R ′ .

We will denote by Ω µ the class of finite relational structures of signature µ and by Ωµ the class of all structures of signature µ. If this causes no confusion, we will denote by 0 the relational structure of signature µ and empty domain. The paper is about Ω µ , and in a lesser extent about Ωµ , quasi-ordered by embeddability. Before entering into details, we first consider some examples.

A collection of examples.

For m = 1, 2, 3, an m-ary relation ρ is unary, binary, ternary, respectively. When needed, we will identify ρ with its characteristic function. Hence, ρ can be viewed as a map from V m to {0, 1}. If m = 1, a unary relation on V can be identified with a subset of V . 1.4.1. Directed graphs, posets, ordered structures. If ρ is a binary relation, the pair R ∶= (V, ρ) is a directed graph; the elements of V are its vertices, pairs (x, x) ∈ ρ are its loops and pairs (x, y) ∈ ρ with x / = y its arcs. If ρ is irreflexive and symmetric, it can be identified with the set E(ρ) of pairs {x, y} such that (x, y) ∈ ρ. The pair (V, E(ρ)) is then an undirected graph, or simply a graph; the elements of E(ρ) are its edges. The complement of an undirected graph

G ∶= (V, E) is the graph G comp ∶= (V, [V ] 2 ∖ E). If R and R ′ are two graphs, a homomorphism from R to R ′ is called a graph homomorphism,
this is a map which maps every edge on an edge or a loop. As indicated in Subsection 1.1, if ρ is reflexive and transitive, R ∶= (V, ρ) is a quasi-ordered set (quoset); if in addition ρ is antisymmetric, R is a (partially) ordered set (poset). If R is a poset or a quoset, we denote by x ≤ y the fact that (x, y) ∈ ρ. A homomorphism f from a poset R into a poset R ′ is an order preserving map, that is, it satisfies f (x) ≤ f (y) for all x ≤ y in V (R). Note that we may view reflexive directed graphs and irreflexive directed graphs as similar objects, but the homomorphisms can be quite different.

Let R ∶= (V, (ρ i ) i∈I ) be a relational structure. When all the relations ρ i , i ∈ I, are binary, we have a binary relational structure, (binary structure for short). We also said that R is a binary structure if all the ρ i 's are binary or unary. If one of the relations ρ i in R ∶= (V, (ρ i ) i∈I ) is a linear order, we have an ordered relational structure. A structure which has the form R ∶= (V, ≤, ρ 1 , . . . , ρ k ), where k is a nonnegative integer, ≤ is a linear order on the set V and each ρ i is a binary relation on V , is an ordered binary structure of type k. Basic examples of ordered binary structures are chains (k = 0), bichains (k = 1 and ρ 1 is a linear order on V ) and ordered directed graphs (k = 1). In the latter case, if ρ 1 is an irreflexive and symmetric binary relation on V , then we just say ordered graphs.

One may ask about the need to consider relations of arity larger than two, several relations on the same set, and particularly infinitely many. Examples speak for themselves and an additional reason to consider relational structures made of infinitely many relations is that they can sometimes provide easy to get counterexamples.

1.4.2. Hypergraphs. A hypergraph is a pair H ∶= (V, E) where E is a collection of subsets of V ; members of V are the vertices of H, whereas members of E are the hyperedges. We denote by V (H), respectively, E(H), the sets of vertices, respectively, hyperedges, of a hypergraph H; we denote by v(H), respectively, e(H), the cardinality of V (H), respectively, E(H). Let ℘(K) be the set of subsets of K. For K a subset of V , the hypergraph induced by H on K is H ↾K = (K, E ∩ ℘(K)). Let h be an integer; the hypergraph H is h-uniform (or h-regular) if all its edges have size h (for instance every graph is a 2-uniform hypergraph). We make the convention that a hypergraph with no hyperedge is h-uniform for every h.

Let H ∶= (V, E) and

H ′ ∶= (V ′ , E ′ ) be two hypergraphs. An isomorphism from H onto H ′ is any bijective map f from V onto V ′ such that the natural extension f to ℘(V ) induces a bijective map from E onto E ′ . A homomorphism from H into H ′ is any map f from V onto V ′ which transforms every hyperedge E ∈ E into a hyperedge f (E) ∈ E ′ . To each h-uniform hypergraph H ∶= (V, E) we may associate the h-ary relational structure R H ∶= (V, ρ E ) where ρ E ∶= {(x 1 , . . . , x h ) ∶ {x 1 , . . . , x h } ∈ E}. Doing so, a map from V to V ′ is a homomorphism of H into H ′ if and only if this is a homomorphism of R H into R H ′ .
Several examples of hypergraphs come from geometry. For instance, Steiner triple systems, which are 3-uniform hypergraphs such that each pair of distinct vertices belongs to exactly one hyperedge. 1.4.3. Words. Let A be a set. Considering A as an alphabet whose members are letters, we write a word α with a mere juxtaposition of its letters as α = a 0 a 1 ⋯a n-1 where a i are letters from A for 0 ≤ i < n. The integer n is the length of the word α and is denoted by |α|. Hence, we identify letters with words of length 1. Let A * be the set of all words on the alphabet A. Once equipped with the concatenation of words, A * is a monoid, whose neutral element is the empty word ◻. In fact, A * is the free monoid generated by A. Ordered with the subword ordering, A * becomes an ordered monoid, this order being defined as follows: if α and β are two elements in A * such that α ∶= a 0 ⋯a n-1 and β ∶= b 0 ⋯b m-1 then α ≤ β if there is an injective and increasing map h from {0, ..., n -1} to {0, ..., m -1} such that for each i, 0 ≤ i ≤ n-1, we have a i = b h(i) . We can code each word α ∶= a 0 ⋯a n-1 by the unary ordered structure L α ∶= ({0, n -1}, ≤, (χ a ) a∈A ) where ≤ is the natural order on {0, . . . , n -1} and χ a is the unary relation χ a ∶= {i ∈ {0, . . . , n -1} ∶ a i = a}. With this coding, a word α is a subword of the word β if and only if L α is embeddable in L β . If A is ordered, the Higman ordering on A * consists of replacing the condition a i = b h(i) in the subword ordering by a i ≤ b h(i) . With this order, A * is the ordered monoid generated by the poset A. If U ∶= ({0, . . . , n -1}, ≤, (ρ i ) i<k ) is an ordered unary structure, we may encode it by the word w U ∶= U (0)⋯U (j)⋯U (n -1), with U (j) ∶= {i < k ∶ j ∈ ρ i }, on the alphabet A ∶= ℘({0, . . . k -1}) ordered by set inclusion. With this encoding, there is an injective homomorphism from U into U ′ if and only if w U ≤ w U ′ for the Higman ordering. Another order -the factor ordering -can be introduced in a similar way. Let C α be obtained from L α by replacing the order relation on {1, . . . , n} by the oriented path C, defined by (x, y)

∈ C if y = x + 1. Then α is a factor of β (that is, β is the concatenation product β ′ ⋅ α ⋅ β ′′ ) if and only if C α embeds in C β .
The study of words and languages (alias sets of words) is at the carrefour of several disciplines (for instance, combinatorics, langage theory and symbolic dynamics). Some important results in language theory use logical techniques applied to words viewed as relational structures (see, for example, the book of Thomas [START_REF] Thomas | Languages, automata, and logic[END_REF]). 1.4.4. Permutations, bichains, permutation graphs. Let n be a nonnegative integer. Let S n be the set of permutations on [n] ∶= {1, . . . , n} and S ∶= ⋃ n∈N S n . An order relation on S is defined as follows: the permutation π of [n] contains the permutation σ of [k] and one writes σ ≤ π if some subsequence of π of length k is order isomorphic to σ. More precisely, σ ≤ π if there exist integers

1 ≤ x 1 < ⋅ ⋅ ⋅ < x k ≤ n such that for 1 ≤ i, j ≤ k ∶ σ(i) < σ(j)
if and only if π(x i ) < π(x j ). For example, π ∶= 391867452 contains σ ∶= 51342, as it can be seen by considering the subsequence 91672 (equals to π(2), π(3), π(5), π(6), π(9)).

A subset

C of S is called hereditary if σ < π ∈ C implies σ ∈ C. Its counting function, referred to as its profile, is φ C (n) ∶= |C ∩ S n |. The Stanley-Wilf conjecture asserted that if C ≠ S then φ C (n) drops from φ S (n) = n! to exponential growth, that is, φ C (n) < c n
for every n for some constant c depending on the class C. The conjecture was proved to be true in 2004 by Marcus and Tardös [START_REF] Marcus | Excluded permutation matrices and the Stanley-Wilf conjecture[END_REF]. Since then a huge literature has appeared about the study of hereditary classes of permutations. An emphasis was put on hereditary classes of finite permutation graphs and a classification of these classes, notably in terms of their profile. The role of the notions of primality and of well-quasi-order has been particularly investigated, see [START_REF] Klazar | Overview of general results in combinatorial enumeration, in Permutation patterns[END_REF][START_REF] Vatter | Permutation classes[END_REF].

Thanks to Cameron [START_REF] Cameron | Homogeneous permutations. Permutation patterns (Otago[END_REF] such a study fits in the framework of the theory of relations. Indeed, a bichain is a binary relational structure B ∶= (V, (L ′ , L ′′ )) made of a set V and two linear orders L ′ and L ′′ on V . To a permutation σ of [n] ∶= {1, . . . , n}, associate the bichain C σ ∶= ([n], ≤, ≤ σ ) where ≤ is the natural order on [n] and ≤ σ the linear order defined by x ≤ σ y if and only if σ(x) ≤ σ(y).

Lemma 1.5. (1) If B ∶= (V, (L ′ , L ′′ )) is a finite bichain, then B is isomorphic to a bichain C σ for a unique σ on [|V |]. (2) If σ and π are two permutations, then σ ≤ π if and only if C σ ≤ C π .
This fact indicates that permutations can be viewed as isomorphism types of bichains. Furthermore, to hereditary classes of permutations correspond hereditary classes of bichains. For a study of some hereditary classes of permutations in terms of bichains, see [START_REF] Cameron | Homogeneous permutations. Permutation patterns (Otago[END_REF][START_REF] Oudrar | Sur l'énumération de structures discrètes: une approche par la théorie des relations[END_REF][START_REF] Oudrar | Profile and hereditary classes of relational structures[END_REF].

Recall that a graph G ∶= (V, E) is a permutation graph if there is a total order ≤ on V and a permutation σ of V such that the edges of G are the pairs {x, y} ∈ [V ] 2 which are reversed by σ. To a bichain R ∶= (V, (≤ ′ , ≤ ′′ )), we may associate the intersection order o(R) ∶= (V, ≤ ′ ∩ ≤ ′′ ) and to o(R) its comparability graph. An order of the form o(R) is called two-dimensional. A graph is the comparability graph of a twodimensional poset if and only if it is also the incomparability graph of a two-dimensional poset [START_REF] Dushnik | Partially ordered sets[END_REF]. If the graph is finite, this amounts to the fact that this is a permutation graph. To a hereditary class of bichains corresponds a hereditary class of two dimensional orders and to a hereditary class of finite two-dimensional orders, a hereditary class of permutation graphs.

Several studies have been made on hereditary classes of finite permutation graphs which are well-quasiordered (for example, [START_REF] Brignall | A counterexample regarding labelled well-quasi-ordering[END_REF], [START_REF] Korpelainen | Boundary properties of well-quasi-ordered sets of graphs[END_REF], [START_REF] Korpelainen | Bipartite induced subgraphs and well-quasi-ordering[END_REF], [124] [125]). 1.4.5. Metric spaces. Recall that a metric space is a set E endowed with a distance function, that is, a map d from E × E in the set R + of nonnegative real numbers such that d(x, y) = 0 if and only if x = y, and for every x, y, z ∈ E, d(x, y) = d(y, x) and d(x, y) ≤ d(x, z) + d(z, y). Part of the study of metric spaces deals with contractions, also called non-expansive maps. A map f from a metric space (E, d) into a metric space

(E ′ , d ′ ) is a contraction if d ′ (f (x), f (y)) ≤ d(x, y) for all x, y ∈ E. Let Q +
* be the set of positive rational numbers. To a metric space (E, d) let us associate the binary structure

R d ∶= (E, (δ r ) r∈Q + * ), where δ r ∶= {(x, y) ∈ E × E ∶ d(x, y) ≤ r}.
Due to the density of Q + * in R + , we trivially have: Lemma 1.6. A map f from a metric space (E, d) to a metric space (E ′ , d ′ ) is a contraction if and only if this is a relational homomorphism from R d to R d ′ . In particular, f is an isometry of metric spaces if and only if f is an isomorphism of binary structures.

In fact, metric spaces, orders and graphs fit in the same frame. See the survey [START_REF] Kabil | Geometric aspects of generalized metric spaces: Relations with graphs, ordered sets and automata in New Trends in Analysis and Geometry[END_REF] and for some specific resuls on ordinary metric spaces viewed as relational structures, see [START_REF] Delhommé | Divisibility of countable metric spaces[END_REF][START_REF] Delhommé | Indivisible ultrametric spaces[END_REF].

1.4.6. Permutation groups. Let G be a group of permutations on a set V ; let G be the topological adherence of G into G(V ), the group of all permutations of V , equipped with the topology induced by the product topology on V V , V being equipped with the discrete topology. Then G is a group, called the closure of G.

If G = G then G is closed.
It is easy to see that: Lemma 1.7. Given a permutation group G on a set V , there is a relational structure R on V such that the two following statements are equivalent. (a) Every isomorphism f from a finite restriction of R onto another extends to an automorphism of R;

(b) Aut(R) = G.
Structures satisfying condition (a) are called homogeneous (or ultrahomogeneous). They are now considered as one of the basic objects in model theory. Their ages are called Fraïssé classes after their characterization by Fraïssé in 1954. If G is a permutation group acting on a set V , the orbital profile of G is the function θ G which counts for each nonnegative integer n the number, possibly infinite, of orbits of the n-element subsets of V . In many cases the coding of G by some relational structure R ∶= (V, (ρ i ) i∈I ) requires I to be infinite, even if θ G (n) is finite for each nonnegative integer n.

Groups for which θ G (n) is always finite are called oligomorphic by Cameron. Their study is a whole subject by itself, see Cameron's book [START_REF] Cameron | Oligomorphic permutation groups[END_REF]. Oligomorphic groups are relevant to model theory due to the following result of Ryll-Nardzewski.

Theorem 1.8. Let G act on a denumerable set V and R be a relational structure such that AutR = G. Then G is oligomorphic if and only if the complete theory of R is ℵ 0 -categorical.

A lot of work has been made on infinite permutation groups and relational structures. Due to the work of Kechris, Pestov and Todorcevic in 2005 [START_REF] Kechris | Fraïssé limits, Ramsey theory, and topological dynamics of automorphism groups[END_REF] the last fifteen years have seen a renewed interest in homogeneous structures, Polish groups and Ramsey theory.

Another motive for considering the orbital profile of a permutation group is provided by this example of Cameron [START_REF] Cameron | Oligomorphic permutation groups[END_REF]. Let k be a nonnegative integer and S be a permutation group of {1, . . . , k}. Let Inv(S) be the algebra of polynomials in the variables X 1 , . . . , X k which are invariant under the action of S (for instance, think about symmetric polynomials). A fundamental result of invariant theory of the nineteen century asserts that this algebra is finitely generated, an upper bound on the degree of the generators being given by Noether. Now, let G be the group of permutations on the product {1, . . . , k} × N in such a way that one can permute the "columns" according to S and permute abitrarily the elements of these columns (this group is the wreath product of S and S(N)). Each orbit of an n-element subset for this group corresponds to a homogeneous polynomial of degree n, hence the generating series associated to the orbital profile of G is the Hilbert series of the algebra Inv(S). This gives a basic example of a permutation group whose orbital profile is bounded by a polynomial. But there are others: a description is given in Falque's thesis [START_REF] Falque | Classification of p-oligomorphic groups; conjectures of Cameron and Macpherson[END_REF]. In Section 6 we consider profiles bounded by polynomials, the algebra of an age invented by Cameron which extends the notion of the algebra of invariants, and we mention the solution by Falque and Thiéry [START_REF] Falque | Macpherson's conjecture holds: the orbit algebra of a permutation group with polynomial profile is finitely generated[END_REF] of a conjecture of Cameron and of Macpherson.

COMPARISON OF RELATIONAL STRUCTURES BY EMBEDDABILITY

In this section we set the fundamental notions of the theory of relations like embeddability, hereditary classes, ages, and bounds. We present the links with universal classes studied in model theory. Hereditary classes which are level-finite play a prominent role in our study. The finiteness of the level can be expressed in terms of profile. For a given hereditary class C of finite structures, the profile of C is the function φ C which counts for each integer n the number (possibly infinite) φ C (n) of elements of the n-th level of C. We devote Section 6 to this notion. A test for its finiteness is given in Lemma 6.7. We start this section with the comparison between isomorphy and equimorphy.

2.1. Embeddability, equimorphy, isomorphy and a glimpse at infinite structures. The embeddability relation defines a quasi-order on the class of all relational structures. This quasi-order is not an order, two relational structures R and R ′ such that R ≤ R ′ and R ′ ≤ R do not need to be equal (they may have different domains). We say that they are equimorphic or that one is a sibling of the other and we set R ≡ R ′ if R and R ′ embed in each other. Trivially, if R and R ′ are isomorphic, then they are equimorphic. Furthermore, if R and R ′ are equimorphic and one is finite, then they must be isomorphic. So the embeddability relation induces an ordering on the set Ω µ of finite relational structures with signature µ (µ ∶= (m i ) i∈I ), these structures being considered up to isomorphism.

Generally, one cannot expect equimorphic structures to necessarily be isomorphic. However, the famous Cantor-Bernstein-Schröder Theorem (see [START_REF] Fraïssé | Theory of relations[END_REF]) states that this is the case for structures in a language with pure equality: if there is an injection from one set into the other and vice-versa, then there is a bijection between these two sets. The same situation occurs in other structures such as vectors spaces, where embeddings are linear injective maps. But as expected, it is generally not the case that equimorphic structures are isomorphic. For example, the chain Q of rational numbers is equimorphic to the chain 1 + Q obtained from Q by adding a least element, but it is not isomorphic to 1 + Q.

The behavior of equimorphy introduces a distinction between the study of infinite structures and the study of the finite ones. The following question was formulated by Thomassé [START_REF] Thomassé | On better-quasi-ordering countable series-parallel orders[END_REF] circa 2000.

Question 1. Let R be a countable relation. Is the number of siblings of R, counted up to isomorphy, 1, ℵ 0 or 2 ℵ 0 ? This question is related to the Bonato and Tardif conjecture [START_REF] Bonato | Mutually embeddable graphs and the tree alternative conjecture[END_REF] of 2006 asserting that for every infinite tree (acyclic and connected graph) T , the number of trees which are siblings of T is either 1 or infinite. Indeed, if a connected graph G is not equimorphic to the graph G ⊕ 1 obtained from G by adding an insolated vertex, then every sibling of G is connected. Hence, for a tree T not equimorphic to T ⊕ 1, the Bonato-Tardif conjecture and Thomassé's question are equivalent (the last one being limited to asking if the number of sibling is one or infinite).

I learned in 2021 from Claude Laflamme that these conjectures and questions were negatively solved by Atsushi Tateno, a student of Alex Scott, who defended his thesis in Oxford in 2008. The author constructed a tree T with only one tree equimorphic but not isomorphic to T . Since that tree T does not embed T ⊕ 1, this answers negatively Thomassé's question. In addition it settles, in the negative, a conjecture of Tyomkin [START_REF] Tyomkyn | A proof of the rooted tree alternative conjecture[END_REF] asserting that if a locally finite tree has a proper sibling, then it has infinitely many siblings unless it is a ray (a one way infinite path). In [START_REF] Abdi | An example of Tateno disproving conjectures of Bonato-Tardif, Thomassé and Tyomkyn[END_REF], the authors revisit the example of Tateno and prove that Thomassé's question has a negative answer for posets.

Bonato and Tardif [START_REF] Bonato | Mutually embeddable graphs and the tree alternative conjecture[END_REF] proved their conjecture for rayless trees. Their result was extended to rayless graphs by Bonato, Bruns, Diestel and Sprüssel [START_REF] Bonato | Twins of rayless graphs[END_REF]. Laflamme, Pouzet and Sauer [START_REF] Laflamme | Invariant subsets of scattered trees. An application to the tree alternative property of Bonato and Tardif[END_REF] in 2017 proved that Bonato-Tardif conjecture holds for trees not embedding the binary tree. Laflamme, Pouzet and Woodrow [START_REF] Laflamme | Equimorphy -the case of chains[END_REF] proved that Thomassé's conjecture holds for chains while Laflamme, Pouzet, Sauer and Woodrow [START_REF] Laflamme | Siblings of an ℵ0-categorical relational structure[END_REF] proved that an ℵ 0 -categorical relational structure has one or infinitely many siblings. Braunfeld and Laskowski improved this result, proving that a countable universal structure has 1, ℵ 0 or 2 ℵ 0 siblings ([21], Corollary 7.3, p.23). Hahn, Pouzet and Woodrow [START_REF] Hahn | Siblings of countable cographs[END_REF] proved that countable cographs have one or infinitely many siblings.

The proofs involve several ingredients, among those, the notion of well-quasi-order plays an important role. To illustrate, we give below some information about Laflamme, Pouzet and Woodrow results on chains [START_REF] Laflamme | Equimorphy -the case of chains[END_REF]. Denote by sib(C) the number of chains, up to isomorphism, that are equimorphic to C.

Theorem 2.1. If C is a chain, then sib(C) is equal to 1 or it is infinite. If C is countable then sib(C) is equal to 1, ℵ 0 or 2 ℵ 0 .
2.1.1. Examples. We give below a sequence of simple properties.

• A chain is well ordered if every nonempty subset has a least element. Ordinals are order types of well ordered chains. As it is well known, sib(C) = 1, for every ordinal or reverse ordinal.

More generally, we have the following result.

Proposition 2.2. If C is a finite sum of ordinals and reverse ordinals, then sib(C) = 1.

The converse is false, for example, sib(ω.ω * + ω 2 ) = 1, where ω * is the dual of the chain ω.

• A chain is scattered if it does not embed the chain of rational numbers. As is well known, a countable chain C is equimorphic to Q if and only if it embeds Q; also sib(C) = 2 ℵ 0 for such a chain.

• A chain C or its order type is a surordinal (a notion due to Jullien and Slater [START_REF] Jullien | Contribution à l'étude des types d'ordres dispersés[END_REF][START_REF] Slater | On a class of order-types generalizing ordinals[END_REF]) if for each x ∈ C the final segment ↑ x generated by x is well ordered. Equivalently, the chain 1 + ω * (obtained by adding a least element to the chain of negative integers) does not embed into C. As it turns out, scattered chains C with few (< 2 ℵ 0 ) siblings are finite sums of surordinals and their reverses. However, there are uncountable dense chains C such that sib(C) = 1, and one such construction is owed to Dushnik and Miller (see [START_REF] Rosenstein | Linear orderings[END_REF]).

The proof of Proposition 2.2 illustrates the notion of wellfoundedness. Let n be the least integer such that C has a decomposition as a sum of n ordinals or reverse ordinals. Choose a decomposition

C ∶= ∑ i<n C i minimal in the sense that if C ∶= ∑ i<n C ′ i is another decomposition with C ′ i ≤ C i for i < n then C ′ i is equimorphic to C i for all i < n.
Since ordinals are well ordered under embeddability, such a decomposition exists. Now consider any chain

C ′ ≡ C. Since C ′ ≤ C, C ′ must be of the form C ′ ∶= ∑ i<n C ′ i with C ′ i ≤ C i for all i < n. Since C ≤ C ′ , the same argument yields that C ∶= ∑ i<n C ′′ i with C ′′ i ≤ C ′ i for all i < n. Since C ′ i ≤ C i we have C ′′ i ≤ C i .
From the minimality of the decomposition of C, we have

C ′′ i ≡ C i hence C ′ i ≡ C i . This yields C ′ i ≃ C i thus C ′ ≃ C.
The proof of Theorem 2.1 uses the fact that the class of surordinals is well-founded under embeddability, a result of Jullien [START_REF] Jullien | Contribution à l'étude des types d'ordres dispersés[END_REF] obtained before Laver [START_REF] Laver | On Fraïssé's order type conjecture[END_REF] proved that the class of countable unions of scattered chains is well-founded and in fact wqo.

2.2.

The poset of finite structures. Let Ω µ be the collection of all finite relational structures with signature µ ∶= (m i ) i∈I . The quasi-order of embeddability between members of Ω µ induces an equivalence relation whose classes are isomorphic types of members of Ω µ . Hence, this quasi-order induces an ordering on the quotient set Ωµ ∶= Ω µ / ≃. For simplicity, we will not hesitate to denote this poset by Ω µ . Not much is known about this poset. Our first problem is: Problem 1. Find a characterization in order theoretic terms of Ω µ .

We state below some simple facts pertaining to this problem. (a) The poset Ω µ decomposes into levels labelled by nonnegative integers, the n-th level consists of the isomorphic types of n-element structures of signature µ (in particular, the level 0 consists of the least element, denoted 0, of Ω µ ). In fact, Ω µ is ranked by the cardinality function, that is, if

T ≤ T ′ ∈ Ω µ , then T is covered by T ′ if and only if |T ′ | = |T | + 1.
Hence, maximal chains between two isomorphic types have the same length. (b) If the level of T ∈ Ω µ is n, that is, T is a n-element structure, then Age(T ), the age of T , has cardinality at most 2 n . (c) The third fact is linked to the notion of semimodularity. Let us recall that a poset P is up-semimodular if the covering relation ≺ satisfies the following property:

(2.1) ∀x, y, z(x ≺ y and x ≺ z and y / = z imply ∃t(y ≺ t and z ≺ t)).

Note the condition y / = z. The poset P is down-semimodular if its dual is up-semimodular. The upsemimodularity is a consequence of the property of amalgamation. We recall that a class C of structures is amalgamable if for every pair of embeddings

f ∶ T → U, f ′ ∶ T → U ′ with T, U, U ′ ∈ C there are T ′ ∈ C and two embeddings g ∶ U → T ′ , g ′ ∶ U ′ → T ′ such that g ○ f = g ′ ○ f ′ .
It is then obvious that if a hereditary subclass (see the definition below) of Ω µ is amalgamable, then it is up-semimodular. Since Ω µ is amalgamable [START_REF] Fraïssé | Theory of relations[END_REF], we obtain the following lemma. Lemma 2.3. As a poset, Ω µ is both down-and up-semimodular.

(d) If the index set I of the signature µ is finite, then the levels of Ω µ are finite, hence Ω µ is countably infinite. If each structure R is made of unary relations, it can be replaced by its domain V (R) labelled by a map f from V (R) into the power set 2 I ; if I is finite, then the structure of Ω µ is not too difficult to grasp. Some details are given in Proposition 2.22 below. A combinatorial property of Ω µ linked to semimodularity is given in Lemma 2.10 below. As for every poset, Ω µ is determined up to isomorphy by the set of its initial segments. These initial segments are hereditary classes that we introduce next.

C if R / ∈ C while R -x ∈ C for every x ∈ V (R).
We denote by Bound(C) the set of bounds of C within the signature µ, these bounds being considered up to isomorphy.

Given a class B of finite structure of Ω µ , set F orb(B) ∶= {R ∈ Ω µ ∶ S / ≤ R for every S ∈ B}. Clearly, F orb(B) is a hereditary class of Ω µ . In fact, every hereditary class of Ω µ is of this form. As a consequence, if the cardinality of the bounds of C is bounded by some integer k and the signature is finite, then there is a polynomial algorithm (in time n k ) to decide whether an n-element relation belongs to C. If the cardinality of bounds is unbounded, the decision problem can be solved in polynomial time, is NP-complete or is intractable.

Examples 2.6. The bounds of the class of finite [START_REF] Abdi | An example of Tateno disproving conjectures of Bonato-Tardif, Thomassé and Tyomkyn[END_REF] chains are one of the following: the empty relation on a singleton; the complete binary relation on a 2-element set; the direct sum of two loops; or, a 3-element oriented cycle with a loop at every vertex; (2) posets are the bounds of chains excluding the direct sum of two loops and including a 3-element directed path with a loop at every vertex;

(3) ordered forests are the bounds of the collection of posets plus the connected 3-element poset with two minimals; (4) posets which can be covered by n chains are those of posets and an (n + 1)-element antichain (by the famous theorem of Dilworth [START_REF] Dilworth | A decomposition theorem for partially ordered sets[END_REF]); (5) posets of jump number at most n are the bounds of chains if n = 0. For n = 1 add to the bounds of posets, the ordinal sum of two antichains; for n = 2 the bounds have been determined by Habib and Rival, they have at most 9 elements; for n ≥ 3 they have at most (n + 2)! elements, El-Zahar, Schmerl [START_REF] El-Zahar | On the size of jump-critical ordered sets[END_REF]. (6) posets of Dushnik-Miller dimension at most n form an infinite set for n ≥ 2. The bounds for n = 2 have been described by Kelly [START_REF] Kelly | The 3-irreducible partially ordered sets[END_REF] and independently by Trotter and Moore [START_REF] Trotter | Some theorems on graphs and posets[END_REF]. The decision problem is polynomial for n = 2. For n ≥ 3, it is NP-complete Yannakakis [START_REF] Yannakakis | The complexity of the partial order dimension problem[END_REF]; [START_REF] Babai | Graph isomorphism in quasipolynomial time[END_REF] graphs are a loop and an arc without any loop; (8) planar graphs form an infinite set: each bound can be described by means of Kuratowski's theorem; (9) comparability graphs form an infinite set: they have been completely described by Gallai [START_REF] Gallai | Transitiv orientbare graphen[END_REF] (the list includes all odd cycles of length at least 5 and the complements of cycles of length at least 5); (10) n-colourable graphs form an infinite set: this is known only for n ≤ 2 (the bounds are the odd cycles);

for n ≥ 3, n-colorability is NP-complete, see Garey and Johnson [START_REF] Garey | The complexity of near-optimal graph coloring[END_REF]; [START_REF] Balogh | Hereditary properties of tournaments[END_REF] perfect graphs are the odd cycles of length at least 5 and their complements. This result by Chudnovsky, Seymour, Robertson and Thomas [START_REF] Chudnovsky | The strong perfect graph theorem[END_REF] answers positively the strong perfect graph conjecture stated by Berge [START_REF] Berge | Les problèmes de flot et de tension[END_REF].

2.5. Ages. The age of a relational structure R is the set Age(R) of restrictions of R to finite subsets of its domain, these restrictions being considered up to isomorphism. Any age is an ideal of Ω µ , that is a nonempty initial segment of Ω µ which is up-directed. One of the most basic results of the theory of relations asserts that the converse (almost) holds. The proof illustrates the role of the embeddability ordering.

Theorem 2.7. (Fraïssé, 1954) Every countable ideal of Ω µ is the age of a countable relational structure.

Proof. Let C be a countable ideal. Then C contains either a greatest element t or a chain C of type ω, Proof. For the first part of the proposition, apply Theorem 2.7. For the second part, define the direct sum of two structures R ∶= (V, (ρ i ) i∈I ) and R ′ ∶= (V ′ , (ρ ′ i ) i∈I ) of signature µ as the relational structure R ⊕ R ′ , defined on the union of their bases supposed disjoint, and made of the relations ρ i ∪ ρ ′ i for i ∈ I. Then the age of the direct sum of all members of Ω µ is equal to Ω µ . This result extends. Lemma 2.10.

say t 0 ≤ t 1 ≤ ⋯ ≤ t n ≤ ⋯, which is cofinal in C (that is, ↓ C = C).
(1) For every R, R ′ ∈ Ω µ the final segment (↑R) ∩ (↑R ′ ) is a union of final segments of the form ↑T where |T | ≤ |R| + |R ′ |.

(2) If some n j of µ ∶= (n i ) i∈I is at least 2, then for every S ∈ Ω µ ∖ {0} the collection Ω µ ∖ ↑S is an age.

Proof. (1). The direct sum T ∶= R ⊕ R ′ of R and R ′ belongs to ↑R ∩ ↑R ′ and |T | ≤ |R| + |R ′ |. If U is any member of ↑R ∩ ↑R ′ select F, F ′ such that U ↾F is isomorphic to R and U ↾F ′ is isomorphic to R ′ . Then T ∶= U ↾F ∪F ′ ∈↑R ∩ ↑R ′ and |T | ≤ |R| + |R ′ |.
(2). Since S is nonempty, Ω µ ∖ ↑ S is nonempty. We then have two possible cases: Case 1. S does not decompose into a direct sum R ⊕ R ′ with R, R ′ nonempty. In this case, Ω µ ∖ ↑ S is the age of the direct sum of all members of Ω µ ∖ ↑ S. Case 2. S decomposes. Let j be such that ρ j is at least binary. In this case, we use a construction based on the negation of a relation. If ρ is a n-ary relation on a set V , we set ¬ρ ∶= V n ∖ ρ. Define the ¬j-direct sum of two structures R ∶= (V, (ρ i ) i∈I ) and R ′ ∶= (V ′ , (ρ ′ i ) i∈I ) of signature µ as the relational structure R ⊕ ¬j R ′ , defined on the union of their bases supposed disjoint, and made of the relations ρ i ∪ ρ ′ i for i ∈ I ∖ {j} and ¬(¬ρ j ∪ ¬ρ ′ j ) for i ∶= j. Then Ω µ ∖ ↑ S is the age of the ¬j-direct sum of all members of Ω µ ∖ ↑ S. Note that if Ω µ consists only of unary relations and S ∶= ({0, 1}, {0}), then Ω∖ ↑ S is not an age. Indeed, the restrictions of S to {0} and to {1} belong to Ω∖ ↑ S but every common extension embeds S.

2.5.1. Ages and ideals. An ideal of Ω µ is representable if it is the age of some relational structure of signature µ. According to Theorem 2.7, countable ideals are representable. But, provided that the signature contains infinitely many integers at least equal to 2, Ω µ contains ideals that are not representable. For example, consider the relational structure D ∶= (R, (δ r ) r∈Q + ) on R, such that for each r ∈ Q, δ r (x, y) = + if and only if |x -y| ≤ r.

Denote by τ (D ↾F ) the isomorphic type of a restriction of D to a finite subset F of R and let A ∶= {τ (D ↾F ) ∶ F ∈ [R] <ω such that δ 1 (x, y) = -for all distinct x, y ∈ F }. Proposition 2.11. A is an ideal included in Age(D) and there is no relational structure M such that Age(M ) = A.

Proof. Let M ∶= (V, (ρ r ) r∈Q + ), be a relational structure with the same arity as D. Set

d M (x, y) ∶= inf{r ∈ Q + ∶ ρ r (x, y) = +}. Then d M is a distance on V provided that τ (M ↾F ) ∈ Age(D) for every F ∈ [V ] <ω . Moreover, a map f with dom(f ) ⊆ V , range(f ) ⊆ R is a local isomorphism from M to D if
and only if this is an isometry. From this observation follows, in particular, that A is up-directed with cardinality 2 ℵ 0 . On the other hand, for every F ∈ [R] <ω such that τ (D ↾F ) ∈ A, the diameter of F is greater than |F | -1. Hence, if Age(M ) ⊆ A, then a ball B(x, r) in (V, d) cannot contain more than 2r + 1 elements. Each ball in V being finite, V is countable, hence Age(M ) / = A. We note that [START_REF] Delhommé | Representation of ideals of relational structures[END_REF] contains several examples, including the one above, of nonrepresentable ideals.

2.6. Universal classes. Classes of models of universal theories provide examples of hereditary classes. In recalling the background, we follow Fraïssé [START_REF] Fraïssé | Cours de Logique Mathématique, tome 1, Relation et formule logique[END_REF], particularly his notion of free operator, except for the notion of universal class (for him universal classes are finitely axiomatizable). To a given signature µ ∶= (m i ) i∈I , associate a language consisting of predicate symbols ρ i of arity m i for each i ∈ I, variables, equality. A first-order sentence (in the language associated with the signature µ) is universal whenever it is logically equivalent to a sentence of the form ∀x 1 ⋯∀x n φ(x 1 , . . . , x n ) where φ(x 1 , . . . , x n ) is a formula built with the variables x 1 , . . . , x n , the logical connectives ¬, ∨, ∧ and predicates =, ρ i , i ∈ I. The sentence is existential if its negation is equivalent to a universal one. A theory is a consistent set T of sentences, closed under deduction and conjunction; the theory T is axiomatized by a set A of sentences if T is the closure of A. If A consists of universal sentences, T is universal. We denote by T ∀ the set of universal sentences which belong to a theory T . The class M od(U ) of models of a set U of universal sentences is said universal too.

Tarski-Vaught theorem (See [START_REF] Fraïssé | Theory of relations[END_REF], section 5.10. Universal classes, p.161, and also [START_REF] Tarski | Contributions to the theory of models[END_REF][START_REF] Vaught | Remarks on universal classes of relational systems[END_REF]) asserts that if the signature µ is finite, each universal class is determined by a hereditary class of finite structures and, more importantly, that class has only finitely many bounds if and only if the former is finitely axiomatizable.

In more precise terms:

Theorem 2.12. Let µ be a finite signature. A class D of structures of signature µ is universal if and only if D is the class of structures R such that Age(R) ⊆ C for some hereditary class C of finite structures. Furthermore, the universal theory of D is finitely axiomatizable if and only if the class C has only finitely many bounds.

This result is based on a simple observation. If R ∶= (V, (ρ i ) i∈I ) is a relational structure on n elements and if I is finite, one may associate an existential sentence θ R ∶= ∃x 1 ⋯∃x n ψ R (x 1 , . . . , x n ) where ψ R (x 1 , . . . , x n ) is a formula with no quantifier that describes R in such a way that a relational structure S satisfies θ R if and only if R embeds in S. With this observation, if C and D are as above, the collection of sentences ¬θ R where R is any bound of C provides an axiomatization of D by universal sentences. If the set of bounds is finite, the axiomatization is finite; as it turns out, the converse holds [START_REF] Fraïssé | Cours de Logique Mathématique, tome 1, Relation et formule logique[END_REF].

Universal sentences provide a syntactical approach to universal classes. A natural object associated to universal sentences is the Boolean algebra made of Boolean combinations of universal sentences, considered up to elementary equivalence. This Boolean algebra is isomorphic to the Boolean algebra generated by the set of M od(φ) where φ runs through universal sentences, or in more concrete terms, generated by the set {F orb(R) ∶ R ∈ Ω µ }. An equivalent object is the Stone space of this algebra. We give a partial description. The key result is the following proposition together with Theorem 1.3. Proposition 2.13. If a hereditary class C of Ω µ is level-finite then:

(1) C is up-closed;

(2) Id(C) is topologically closed in I(C);

(3) Id(C) with the topology induced by the product topology is the Stone space dual of the Boolean algebra generated by up(C).

Proof. [START_REF] Abdi | An example of Tateno disproving conjectures of Bonato-Tardif, Thomassé and Tyomkyn[END_REF]. Since C is level-finite it follows from (1) of Lemma 2.10 that C is up-closed.

(2). With (1) and the fact that C has a least element, Proposition 1.2 asserts that the set Id(C) is a closed subset of ℘(V ), hence of I(C).

(3). Apply Theorem 1.3.

If µ is finite, Ω µ is level-finite and thus, (3) applies. Since up(Ω µ ) is (dually) isomorphic to {F orb(R) ∶ R ∈ Ω µ } one gets Proposition 2.9 of [START_REF] Bekkali | Incidence structures and Stone-Priestley duality[END_REF].

Theorem 2.14. If µ is finite then the set Id(Ω µ ), equipped with the topology induced by the product topology, is the dual of the Boolean algebra made of Boolean combinations of universal sentences, considered up to logical equivalence. Thus, from Theorem 2.14 the tail algebra T ailalg(Ω µ ) provides an alternative description of the algebra made of Boolean combinations of universal sentences (see [150] [162] for more details on this correspondence).

Instead of Boolean algebras, one may prefer to consider lattices. Proposition 2.13 above yields:

Corollary 2.15. If µ is finite, then the set of hereditary classes of Ω µ having finitely many bounds is closed under finite intersections and finite unions, hence is a distributive lattice dually isomorphic to T aillat(Ω µ ).

2.6.1. Reducts. If the signature is infinite, one needs the notion of reduct. Let µ ∶= (m i ) i∈I be a family of nonnegative integers. Let I ′ ⊆ I and denote the restriction of µ to I ′ by µ ↾I ′ . For a µ-ary relational structure R ∶= (V, (ρ i ) i∈I ), the I ′ -reduct of R is the µ ↾I ′ -ary relational structure R ↾I ′ ∶= (V, (ρ i ) i∈I ′ ). The reduct-closure of a class C of relational structures with signature µ is the class Ĉ of R with signature µ such that for each finite I ′ ⊆ I there is some

R ′ ∈ C such that R ↾I ′ = R ′↾I ′ . A class C is said to be reduct-closed if Ĉ = C. If the signature µ is finite, every class is reduct-closed.
Proposition 2.16. Let C be a hereditary class of finite structures.

(

) If C is level-finite, then C is reduct-closed. ( 1
) If C is reduct-closed, then the class C of structures R such that Age(R) ⊆ C is reduct-closed. 2
Proof. [START_REF] Abdi | An example of Tateno disproving conjectures of Bonato-Tardif, Thomassé and Tyomkyn[END_REF]. We prove that Ĉ = C. Let S ∶= (V (S), (ρ i ) i∈I ) ∈ Ĉ. For every finite subset I ′ of I there is some member S(I ′ ) ∈ C such that S ↾I ′ = S(I ′ ) ↾I ′ . Let n ∶= |V (S)|. Since C is level-finite, there are only finitely many S(I ′ ) ∈ C, hence there is some finite I ′ ⊆ I such that S(I ′ ) = S(I ′′ ) for every finite I ′′ containing I ′ . This implies that S = S(I ′ ), hence S ∈ C as required.

(2). We prove that

Ĉ = C. Let R ∶= (V (R), (ρ i ) i∈I ) ∈ Ĉ. We need to prove that R ∈ C, that is R ↾V ′ ∈ C for every finite V ′ ⊆ V (R).
Since C is reduct-closed, this means that for every finite subset I ′ ⊆ I there is some

S(I ′ ) ∈ C with domain V ′ such that S(I ′ ) = R ↾V ′ . Since R ∈ Ĉ there is some R(I ′ ) ∈ C such that R ↾I ′ = R(I ′ ) ↾I ′ . It suffices to set S(I ′ ) ∶= R(I ′ ) ↾I ′ ↾V ′ .
From this result, it follows that if we use the compactness theorem of first-order logic within the class Ĉ associated to a reduct-closed hereditary class C, the result we obtain belongs to Ĉ. Several examples illustrate this observation. Provided that we consider only reduct-closed classes of finite structures, the above characterization of universal classes given in Theorem 2.12 holds. Also, note that independently of the cardinality of the signature, every ideal that consists of the collection of finite models of a set of universal sentences is the age of some relational structure. For another illustration, see Proposition 5.24.

2.6.2. A construction of universal classes. We reproduce here some properties of free interpretability developed by Fraïssé in [START_REF] Fraïssé | Cours de Logique Mathématique, tome 1, Relation et formule logique[END_REF][START_REF] Fraïssé | Theory of relations[END_REF]. Let µ and ν be two signatures (possibly distinct). A µ-ary relational structure S is freely interpretable in a ν-ary relational structure R if R and S have the same base and every local isomorphism of R is a local isomorphism of S. A map P from the class of µ-ary relational structures into the class of ν-ary relational structures is a free operator if P(R) is freely interpretable by R, for every µ-ary relational structure R. Lemma 2.17. A map P from the class of µ-ary relational structures into the class of ν-ary relational structures is a free operator if and only if for every µ-ary relational structure R:

(1) P(R) has the same domain as R;

(2) P(R↾ A) = P(R)↾ A; and

(3) every automorphism of R is an automorphism of P(R).

Lemma 2.18. Let R and S be two relational structures with the same base V and signatures µ and ν, respectively. Then:

(1) S is freely interpretable by R if and only if there is some free operator P such that P(R) = S;

(2) if µ is finite, P(R) = S for some free operator, and S ′ is such that Age(S ′ ) ⊆ Age(S), then there is an R ′ such that P(R ′ ) = S ′ and Age(R ′ ) ⊆ Age(R).

Proof.

(1) This is a basic result, see [START_REF] Fraïssé | Theory of relations[END_REF], 9.3.1, pp. 247-248.

(2) The proof illustrates the importance of compactness. Let µ ∶= (m i ) i∈I , V ′ be the domain of S ′ and U µ (V ′ ) ∶= Π i∈I 2 (V ′ ) m i . By definition, U µ (V ′ ) is the set of µ-ary relational structures R ′ with domain V ′ . This set has a topological structure for which it is compact. (Equip 2 ∶= {0, 1} with the discrete topology and the set 2 (V ′ ) m i with the product topology; since U µ (V ′ ) is a product, we can equip it with the product topology.) For every finite subset F of V ′ , let U F be the subset of U µ (V ′ ) made of µ-ary relational structures R ′ on V ′ such that P(R ′ ↾F ) = S ′ ↾F . It can be checked that the sets of the form U F are closed and nonempty, hence their intersection is nonempty. Then one checks that any member of this intersection yields the required R ′ .

Illustrations are given by the following theorem and by Lemma 5.36.

Theorem 2.19. The image of a universal class over a finite language by a free operator is a universal class.

Proof. Let C be a universal class made of µ-ary relational structures and P be a free operator that maps each µ-ary relational structure to a ν-ary relational structure. Let D ∶= P(C). Due to (2) of Lemma 2.18 above, D is a hereditary class. To conclude that D is a universal class, it suffices to prove that if S is a ν-ary relational structure on a set V , then there is some R ∈ C such that P(R) = S provided that for every finite F ⊆ V , S↾ F ∈ P(C). This is just (2) of Lemma 2.18.

Here are several universal classes obtained by this process.

Examples 2.20.

(1) The class of ordered sets of order-dimension at most n: to (V, ≤ 1 , . . . , ≤ n ) associate (V, ⋂ i≤n ≤ i ).

(2) The class of posets coverable by at most n chains, n ∈ N.

(3) The class of comparability graphs.

(4) The class of n-colorable graphs, for each n ∈ N. This is a famous result of de Bruijn-Erdös [START_REF] De Bruijn | A color problem for infinite graphs and a problem in the theory of relations[END_REF].

Note that the hypothesis that the signature is finite in Lemma 2.18 (2) is needed. Indeed, if a graph G has chromatic number ℵ 0 , it is freely interpretable by G with ℵ 0 unary relations defining a covering of G in independent sets, a graph G ′ with a smaller age than G may have a larger chromatic number.

2.7.

The number of hereditary classes of finite structures. We may ask how many hereditary classes of finite structures there are. The first result that I saw in this direction was in Mal'cev [START_REF] Mal'cev | The metamathematics of algebraic systems[END_REF].

Lemma 2.21. There are uncountably many hereditary classes of finite graphs.

Proof. Let C n be the n-vertex cycle. The family of these cycles for n ≥ 3 forms an antichain for embeddability since

C n is not embeddable in C m for m / = n. For each subset X of the set N of nonnegative integers, the collection C X of graphs G embeddable into some C n+3 , for some n ∈ X, is by construction a hereditary class. Now, if X / = X ′ , then C X / = C X ′ .
This provides as many classes as subsets of N. Alternatively, for each subset X of the powerset ℘(N), the collection C ¬X of graphs G in which no C n+3 , n ∈ X, is embeddable is a hereditary class and in fact a universal class. The trace of C ¬X on the class G of finite graphs is a hereditary subclass. This class is up-directed, since it contains the direct sum of any of two of its members, thus it is an age. Hence, we get a continuum number of ages.

Building on Lemma 2.21, we get: Proposition 2.22. Let µ ∶= (m i ) i∈I be a signature and κ ∶= max{ℵ 0 , 2 |I| }. Then |Ω µ | = κ and, as a poset, Ω µ is embeddable in [κ] <ω , the collection of finite subsets of κ ordered by set inclusion. If I is infinite or m j ≥ 2, for some j ∈ I, then Ω µ is equimorphic to [κ] <ω . Otherwise, Ω µ is equimorphic to the power set

N 2 |I| ordered componentwise. Proof. First, let us see that |Ω µ | = κ. Suppose that I is infinite.
Then Ω µ has the same cardinality as 2 |I| . Indeed, a relational structure of signature µ and domain V is an element of Π i∈I 2 V m i . If V is finite, with n elements, this gives 2 |I| structures; since each isomorphy class is finite, the number of isomorphic types is 2 |I| . Since Ω µ is the union of countably many levels and each level (except level 0) has cardinality 2 |I| , Ω µ has cardinality 2 |I| . Suppose that I is finite. In this case, κ = ℵ 0 , each level of Ω µ is finite, hence Ω µ is countably infinite. The embeddability of Ω µ in [κ] <ω follows from the fact that for each τ ∈ Ω µ , the subset

↓ τ ∶= {τ ′ ∈ Ω µ ∶ τ ′ ≤ τ } of Ω µ is finite (it has cardinality at most 2 |τ |
, where τ is the number of elements of the domain of any representative of τ ). Indeed, as for every poset, the map ↓ defines an embedding of Ω µ in [Ω µ ] <ω . This proves the first assertion. Now, suppose that I is infinite. Then κ = 2 |I| . Since the cardinality of the set of members of Ω µ on a one element set is 2 |I| = κ there is a set V of cardinality κ and a relational structure R of signature µ based on V such that all its restrictions to singletons are pairwise non-isomorphic. If we associate to each finite subset F of V the restriction R ↾F we get an embedding of [V ] <ω in Ω µ . Hence, Ω µ is equimorphic to [Ω µ ] <ω . If I is finite, we suppose first that µ consists only of the integer 2. As in the proof of Lemma 2.21, let (C n+3 ) n∈N be the family of cycles of length at least 3. To each finite subset F of N associate ⊕ n∈F C n+3 , the direct sum of the C n+3 for n ∈ F and observe that this defines an embedding from [N] <ω in the collection G of finite graphs, hence in Ω µ . Suppose next that there is some j ∈ I such that m j ≥ 2. To each finite graph G ∶= (V, E) associate the relational structure R G ∶= (V, (ρ i ) ∈I ) where ρ i ∶= ∅ if i / = j and

ρ j ∶= {(x 1 , . . . , x m j ) ∈ V m j ∶ (x 1 , x 2 ) ∈ E} if i = j. This defines an embedding of G in Ω µ . Since [N] <ω embeds in G, it embeds in Ω µ .
Hence, these two posets are equimorphic.

Finally, suppose I finite, I ∶= {1, . . . , k}, and that m i = 1 for i ∈ I. We may represent each finite R ∈ Ω µ by a finite multiset over 2 k . Indeed to R ∶= (V, (ρ i ) i=1,...,k ) we may associate (V, χ R ) where χ R is the map from V into 2 k defined by setting χ R (x) ∶= (ρ i (x)) i=1,...,k . Next, to (V, χ R ) we may associate its frequency F R ∶= (n s ) s∈2 k where n s is the number of elements x of V such that χ R (x) = s. These frequencies are members of the power P ∶= N 2 k that we can order componentwise. One can check that R ≤ R ′ if and only if

F R ≤ F ′ R .
Hence, Ω µ is equimorphic to P . The result follows. As mentioned in Subsection 1.2, if P is a poset, then the poset I(P ) of initial segments of P ordered by set inclusion is order isomorphic to Id(I <ω (P )). Applying this to the set N of nonnegative integers ordered by the equality relation, we get that the powerset ℘(N) ordered by set inclusion is isomorphic to Id([N] <ω ). This added to Proposition 2.22 yields: Lemma 2.23. If the maximum of the signature µ ∶= (m i ) i∈I is at least 2 and I is finite, then the set Id(Ω µ ) of ages of Ω µ , ordered by set inclusion, is equimorphic to the powerset ℘(N) ordered by set inclusion.

Proof. According to Proposition 2.22, Ω µ is equimorphic to [N] <ω . It follows that Id(Ω µ ) is equimorphic to Id([N] <ω ) which in turn is isomorphic to ℘(N).
Since I is finite, the ages of Ω µ constitute ideals which completes the proof.

Dickson's lemma asserts that a finite cartesian product of copies of the chain N ordered componentwise contains no infinite antichain. Hence, from Proposition 2.22, we get: Corollary 2.24. Let µ ∶= (m i ) i∈I be a signature. Then Ω µ has no infinite antichain if and only if I is finite and m i = 1 for every i ∈ I.

Thus, either Ω µ contains no infinite antichain, in which case Id(Ω µ ) is countable, or Id(Ω µ ) is uncountable. Does this dichotomy extend to hereditary classes? With this question, well-quasi-ordering enters into the study of hereditary classes of finite structures.

TYPOLOGY OF HEREDITARY CLASSES AND AGES

In this section and in Section 5, we present the results of attempts to classify hereditary classes and ages based on order and topological notions.

One could classify hereditary classes according to their proper subclasses. With this idea, the simplest classes are those which contain finitely many proper subclasses, hence these classes are finite. At the next level, there are the classes which contain infinitely many proper subclasses, but every proper subclass contains only finitely many. That is, such classes are infinite but proper subclasses are finite (we call them Jónsson classes). It is a simple exercise based on Ramsey's theorem to prove that among hereditary classes of finite graphs there are only two such classes: the class of finite cliques and the class of their complements. Pursuing this idea further, one would like to attach a kind of rank to each class, preferably an ordinal. If we do this, it turns out that a class has a rank if and only if the set of its proper subclasses ordered by set inclusion is well-founded. This latter condition amounts to the class being well-quasi-ordered. This allows us to apply to hereditary classes results of the rich literature concerning well-quasi-ordered posets by now well-known [START_REF]The theory of well-quasi-ordering: a frequently discovered concept[END_REF], notably to construct wqo hereditary classes. Above all, we are lead to classifying ages by the same process. After recalling the notion of better-quasi-order (bqo) invented by Nash-Williams [START_REF] St | On well-quasi-ordering infinite trees[END_REF][START_REF] St | On well-quasi-ordering finite trees[END_REF] and the Nash-Williams conjecture, we are lead to consider four types of (potentially distinct) ages: the ages which are better-quasi-ordered, the well-quasi-ordered ages, the ages for which the collection of subages is countable, and those for which the collection of subages is well-founded. If the signature is finite, it is possible that wqo ages are necessarily bqo. Otherwise, there are counterexamples.

A central question is to know whether or not an age is wqo whenever the collection of subages is wellfounded. This old question remains unanswered. It is discussed in Problems 4 and 5 and in Conjecture 1. A parallel with the study of join semilattices is mentioned. The notions of kernel and generation are introduced. Applications to a topological study of the collection of subages of a hereditary class (Theorems 3.40 and 3.41) are given; the Vietoris topology and the Vietoris rank are mentioned. By way of conclusion, we observe that the problems and conjecture mentioned above have a positive answer for ages of posets coverable by two chains.

3.1.

Well-quasi-ordering. A poset P is well-quasi-ordered (wqo for short) if every nonempty subset contains finitely many minimal elements (this number being non-zero). Equivalently, P is wqo if it contains no infinite descending chain and no infinite antichain. A final segment F of a poset P is finitely generated if for some finite subset K of P , F =↑ K.

We recall a result by Higman [START_REF] Higman | Ordering by divisibility in abstract algebras[END_REF]:

Theorem 3.1. The following properties are equivalent for a poset P : (i) P is wqo;

(ii) every final segment F of P is finitely generated; (iii) the set I(P ) of initial segments of P , ordered by set inclusion, is well-founded.

Condition (iii) in Theorem 3.1 allows to do induction on the collection of initial segments. An example of easy induction is given by the following result. Theorem 3.2. A wqo poset is a finite union of ideals.

Proof. If this is not true, take a wqo poset P for which this is not true. Since I(P ) is well founded, it contains a minimal initial segment I which is not a finite union of ideals. Due to the minimality condition, this I cannot be the union of two proper initial segments. But then it is up-directed. Indeed, if x, y ∈ I have no upper bound in I then I is the union of two proper initial segments I∖ ↑ x and I∖ ↑ y. Since I cannot be empty (the empty set is an empty union of ideals), this is an ideal. This is a contradiction.

The previous two results remind us of the characterization of noetherian rings and the fact that in a noetherian ring every ideal is a finite intersection of irreducible ideals.

The wqo condition is not necessary in Theorem 3.2; the fact that there is no infinite antichain suffices (take a well-founded cofinal subset, it will be wqo). A necessary and sufficient condition for a poset to be a finite union of ideals is due to [START_REF] Erdös | On families of mutually exclusive sets[END_REF] [START_REF] Erdös | On families of mutually exclusive sets[END_REF]. Simple, beautiful, this is one of the first min-max results in the theory of ordered sets. We will state and prove it.

Let us say that two elements of a poset P are incompatible if they do not have a common upper bound and let us say that a subset A of P is inconsistent if no pair of distinct elements of A are compatible. Theorem 3.3. If a poset P contains no infinite inconsistent set, the size of inconsistent sets is bounded by a nonnegative integer n and the least upper bound is the minimum number of ideals needed to cover P .

Proof. Let F be the set of x ∈ P such that the final segment ↑ x is up-directed. This is a final segment. Among the inconsistent subsets of F , pick a maximal one with respect to set inclusion, say A. From the maximality of A, we deduce that the union of the sets I x ∶=↓ (↑ x) for x ∈ A will cover I ∶=↓ F . If I does not cover P , build a copy of the binary tree T ∶= 2 <ω in P ∖ I in such a way that at every stage, the images of the sequences s0 and s1 (obtained by adding 0 and 1 to a sequence s ∈ T ) are two incompatible elements above the image of s. Then, the images of the sequences 11 . . . 10 form an inconsistent set.

The most useful trick to prove that a poset is wqo is the minimal bad sequence technique discovered by Nash-Williams [START_REF] St | On well-quasi-ordering finite trees[END_REF], and the most important tools to construct wqo sets are due to Higman and Kruskal ([94,[START_REF]Well-quasi-ordering, the tree theorem, and Vazsonyi's conjecture[END_REF]). We present here the minimal bad sequence technique (in a simple form) and will present the Higman and Kruskal theorems in Subsection 3.3.

Let P be a poset; a sequence (q n ) n<ω of elements of P is bad if q n / ≤ q m for n < m. This sequence is minimally bad if there is no bad sequence (q ′ n ) n<ω and some m < ω such that q ′ n = q n for n < m and q ′ m < q m . Nash-Williams's lemma asserts that: Lemma 3.4. If a poset P is well-founded but not wqo then it contains a minimally bad sequence.

Proof. Let q 0 ∈ P be minimal such that there is some bad sequence (q ′ n ) n<ω with q ′ 0 = q 0 . Supposing q n defined for n < m such that there is some bad sequence (q ′ n ) n<ω with q ′ n = q n for n < m. Among the m-th terms of these sequence, select q ′ m be minimal and set q m = q ′ m . This lemma is tricky and powerful. A fact, inspired from the theory of rings, simpler to conceive and as powerful is the following.

Let P be a poset. Suppose that P contains a final segment F which is not finitely generated. Then it follows from Zorn's lemma that F is contained in a final segment maximal, with respect to set inclusion, with the same property. If F is such a final segment then P ∖ F is wqo. Indeed, if F ′ is any final segment of P ∖ F then F ∪ F ′ is a final segment of P . This final segment is finitely generated iff F ∪ F ′ / = F , that is, F ′ is nonempty or equivalently, F ′ is nonempty and finitely generated. If P is well-founded, then the set A ∶= min( F ) is an infinite antichain of P and every element of P ∖ F , if any, is below almost every element of A. With that, we get: Lemma 3.5. If a level-finite and up-closed poset P contains an infinite antichain then it contains a wqo ideal I such that min(P ∖ I) is an infinite antichain.

This was observed in several works; see, for instance, the comment after Corollary 4.6 and Lemma 4.7 of [START_REF] Pouzet | The profile of relations[END_REF].

One of the most important results about wqo is the de Jongh-Parikh Theorem (see [START_REF] Wolk | Partially well-ordered sets and partial ordinals[END_REF] for the first part, and [START_REF] De Jongh | Well-partial orderings and hierarchies[END_REF] for the second).

Theorem 3.6. If a poset P is wqo then all the linear extensions of P are well ordered and there is one having the largest possible order type.

This largest order type, denoted o(P ), is the ordinal length of P . If a poset P is wqo, then since the set I(P ) of initial segments of P , ordered by set inclusion, is well-founded, then we may decompose it into levels via the height function. An equivalent formulation of de Jongh-Parikh's theorem was given in [START_REF] Pouzet | Hereditary classes of ordered sets coverable by two chains[END_REF].

(A * ) = ω ω p-1 [50].
The ordinal length of several other wqo posets have been computed. For example, the habilitation of D.Schmidt [START_REF] Schmidt | Well-partial orderings and their maximal order types[END_REF] contains the extension to wqo alphabets of the above result and several results on classes of labelled trees. The ordinal length of the class of finite trees was computed by Rathjen and Weiermann [START_REF] Rathjen | Proof-theoretic investigations on Kruskal's theorem[END_REF] (here the trees are graphs and the embeddings are topological embeddings).

3.2.

Well quasi-ordered hereditary classes. Since in Ω µ each principal initial segment is finite, hereditary subclasses C of Ω µ with no infinite antichains are wqo. In particular, if the maximum of the signature µ is 1 and the number of unary symbols is finite, Ω µ is wqo (Corollary 2.24). This observation leads us to some examples of hereditary classes, but in fact, very few. If the maximum of the arity is exactly 2, the situation is completely different. As we will show in Section 7 there are 2 ℵ 0 hereditary subclasses of Ω µ which are wqo. Except for very few, given right below, the construction of examples requires work, so we prefer to present first the general properties of wqo hereditary classes.

Examples 3.9. The simplest example of a wqo hereditary class of finite structures is Chain, the class of finite chains. Indeed it is totally ordered by embeddability, thus has order type ω. What is trivial for finite chains is not for the infinite; the extension of the wqo character to the class of countable chains was the famous conjecture of Fraïssé, solved by Laver. The fact that Chain is totally ordered by embeddability was extended by Fraïssé to chainable structures; some properties of the class of these structures are in Subsection 5.1. Beyond Chain, there are two typical examples: namely the collection of direct sums of finite chains and the collection of direct sums of paths. The fact that these classes are wqo follows immediately from a theorem of Higman given in Subsection 3.3. But such a deep result is not necessary here. Indeed, let I <ω (ω × ω), be the set, ordered by set inclusion, of finitely generated subsets of the direct product of ω by ω. This poset is wqo. It is not too difficult to prove that the collection ⊕ Chain of direct sums of finite chains, quasi-ordered by embeddability, yields a poset isomorphic to this poset. As a poset, the collection of direct sums of paths, quasi-ordered by embeddability, yields a poset extending the previous one, thus is wqo. Note that while these two posets are not isomorphic they have the same profile, namely the partition function.

Structural properties of wqo hereditary classes start with the translation of Theorem 3.1 in terms of hereditary classes. We see that (c) implies (d) because ℘(N) ordered by set inclusion is not well-founded. Finally, we show (d) implies (a). Indeed, if C contains an infinite antichain, say A, let (G n ) n∈N be an enumeration of A. The map φ ∶ ℘(N) → I(A) defined by setting φ(I) ∶= {G ∶ G ≤ G n for some n ∈ I} for each subset I of N is an embedding, demonstrating the negation of (d).

Note that every wqo hereditary subclass C of Ω µ is countable, even when µ is infinite (indeed, each level, being an antichain, is finite). Hence, wqo ideals are ages. From Theorem 3.2 (or Theorem 3.3) we obtain: Corollary 3.11. Every wqo hereditary class C of finite structures is a finite union of wqo ages.

Since a hereditary class which is level-finite is necessarily up-closed (cf. (1) of Proposition 2.13), Lemma 3.5 yields the following result (cf. [START_REF] Pouzet | The profile of relations[END_REF], Lemma 4.7 and [155], 3.9, p. 329): Lemma 3.12. A level-finite hereditary class C is wqo if and only if every age properly included into C has only finitely many bounds belonging to C.

We illustrate this result with Corollary 4.11 and Theorem 5.51.

3.2.1.

Ordinal length of well-quasi-ordered hereditary classes and height of an age. Let C be a hereditary class of finite structures. If C is wqo, then according to de Jongh-Parikh's result, C has an ordinal length. Since C is countable this ordinal length is a countable ordinal. Since, in this case, Id(C) is well-founded, we can compute its height (defined by formula (1.1) by setting

h(A, Id(C)) ∶= sup{h(A ′ , Id(C)) + 1 ∶ A ′ ∈ Id(C) and A ′ ⊂ A}, and h(Id(C)) ∶= sup{h(A) + 1 ∶ A ∈ Id(C)}.
This height, that we also denote H(C), is defined provided that Id(C) is well-founded. One of the main unsolved questions is to know whether or not an age A is wqo whenever Id(A) is well-founded (see Problem 5). Despite that, we have the following. Theorem 3.13. Let C be a level-finite hereditary class of finite structure. If Id(C) is well-founded then H(C) is at most countable.

This result is a consequence of general results of descriptive set-theory [START_REF] Moschovakis | Descriptive set theory[END_REF]. In our case, we use the following result (for a proof, see [START_REF] Pouzet | Sandwiches of ages[END_REF]). Proposition 3.14. Let V be a countable set and F be a family of subsets of V such that F is closed in the space ℘(V ), endowed with the pointwise convergence topology, and F is well-founded under set inclusion. Then h(F) < ω 1 .

Problems 2.

(1) Is every countable ordinal attained by the ordinal length of some wqo hereditary class of finite structures with a given finite signature? (2) Is there a largest countable ordinal depending upon the signature? (3) Is there a countable ordinal which bounds the ordinal length of wqo hereditary classes of graphs?

The ordinal length of the class of finite (ordered) forests is the ordinal ϵ 0 . The ordinal length of the class of finite series-parallel posets is the ordinal Γ 0 [START_REF] Pouzet | The order type of the collection of finite series-parallel posets[END_REF]. The ordinal length of the class of finite cographs is at most Γ 0 ; we do not know if the equality holds.

If we allow unbounded signature, the answer to the first problem above is negative: every ordinal below ω 1 , the first uncountable ordinal, can be attained. Indeed, let α be a countable ordinal; the poset P intersection of two orders, one of order type ω, the other of order type α, is embeddable into [ω] <ω . According to Corollary 1.4 of [START_REF] Delhommé | Length of an intersection[END_REF], o(P ) = α. Now, use a construction presented in Subsection 3.4.1. Let V ∶= P ∪ A, where A is an infinite set disjoint from P . Let F ∶= {↓x ∶ x ∈ P }, M F as defined in 3.4.1 and A ∶= Age(M F ). According to Lemma 3.26 below, A is the image of the direct product F <ω × ω by an order preserving map, hence it is wqo (in fact bqo). Since it embeds F, that is P , its ordinal length is at least α.

There is a relationship between the ordinal length of a poset P and the height of ideals of P . For example, if P is wqo and up-directed then o(P ) = h(P, I(P )) ≤ ω H(P ) where H(P ) ∶= h(P, Id(P ) and Id(P ) is the set of ideals of P [START_REF] Zaguia | Chaînes d'idéaux et de sections initiales d'un ensemble ordonné[END_REF].

The computation for ages yields a better upper-bound.

Theorem 3.15. If Id(Age(R)) is wqo then o(Age(R)) = ω α ⋅ q, where α is such that ω ⋅ α ≤ H(Age(R)) < ω ⋅ (α + 1)
and q is the number of ages included into Age(R) whose height is between ω ⋅ α and ω ⋅ (α + 1).

This result mentioned in [START_REF] Pouzet | The profile of relations[END_REF] (cf. Theorem 4.12) is included in [START_REF] Sobrani | Sur les âges de relations et quelques aspects homologiques des constructions D+M[END_REF] and a forthcoming paper [START_REF] Pouzet | Ordinal invariant of an age[END_REF]. With Example 3.8, it yields:

Example 3.16. Let R ∶= (Q, U 1 , . . . , . . . U p ) be the chain of rational numbers, colored with p colors U 1 , . . . , U p , in a dense way. Then, H(Age(R)) = ω p , while o(Age(R)) = ω ω p-1 .
In Subsection 3.6 and Section 5, we look at ages of height less than ω 2 .

Construction of wqo hereditary classes.

A theorem of Kruskal [START_REF]Well-quasi-ordering, the tree theorem, and Vazsonyi's conjecture[END_REF] asserts that finite (ordered) trees labelled by a wqo form a wqo. A weaker version, expressed in terms of algebras quasi-ordered by divisibility, is due to Higman [START_REF] Higman | Ordering by divisibility in abstract algebras[END_REF]. A simpler version, often cited as Higman's theorem, asserts that an algebra with only finitely many operations and quasi-ordered by divisibility is wqo provided that some generating set is wqo (see Cohn [START_REF] Cohn | Universal algebra[END_REF], Theorem 2.9). The complete version, with a wqo set of operations, is given in Exercise 10 of Chapter III [START_REF] Cohn | Universal algebra[END_REF]. The set of finite structured trees (ordered trees with a linear order on the successors of each node) labelled by a set can be viewed as an ordered algebra. Hence, the complete version is equivalent to Kruskal's theorem. In the sequel, we present the simpler version of Higman's theorem, because it is very easy to use.

An (abstract) algebra is a pair A ∶= (A, F ) where F is a set of finitary operations on A, each one being a function f of A a(f ) into A, where a(f ) is a nonnegative integer, called the arity of f . A subset X of A generates A if A is the sole subalgebra of A containing X. Supposing that F is finite, a quasi-order ≤ on A is a divisibility quasi-order if:

(

1) each f ∈ F is order preserving, that is, f (u) ≤ f (v) for all u ≤ v in A a(f ) equipped with the product quasi-order; (2) each f ∈ F is extensive, that is, u i ≤ f (u) for every f ∈ F , u ∶= (u 0 , . . . , u i , . . . u a(f )-1 ) ∈ A a(f ) ,
i < a(f ). Higman's theorem reads as follows:

Theorem 3.17. An algebra A quasi-ordered by divisibility is wqo provided that some generator is wqo.

If A is an ordered set, the ordered monoid A * is generated by the set of words of length 1; this set is order isomorphic to A. The Higman ordering on A * is a divisibility ordering, hence, from Theorem 3.17 above: Theorem 3.18. If A is wqo, the ordered monoid A * equipped with the Higman ordering is wqo.

The next consequence of Higman's theorem is about cographs and series-parallel posets.

3.3.1.

Cographs and series-parallel posets. Let us recall that a (loopless undirected) graph is a cograph if it does not embed a path on four vertices. It was proved by Sumner [192] that if G is a finite cograph then either G or its complement G comp is disconnected. It follows that the set of isomorphic type of finite cographs coincides with the set G consisting of the empty graph and the one-vertex graph and the direct sum G ⊕ G ′ and the complete sum G + G ′ of any two of its members. Equipped with the operations of direct and complete sums, and ordered by embeddability, G is a algebra quasi-ordered by divisibility. Since G is generated by a two-element set, it is wqo.

A poset P is series-parallel if it does not embed a N , the poset made of four elements a, b, c, d such that a < b and c < b, d are the only comparabilities. This amounts to the fact that the comparability graph of P is a cograph. Due to the recursive construction of finite cographs recalled above, the set of isomorphic type of finite series-parallel posets coincides with the set N consisting of the empty poset and the one-vertex poset and the direct sum P ⊕ P ′ and the ordinal sum P + P ′ of any two of its members. Equipped with the operations of direct and complete sums, and ordered by embeddability, N is a algebra quasi-ordered by divisibility. Since N is generated by a two-element set, it is wqo.

This yields the beautiful result of Damaschke [START_REF] Damaschke | Induced subgraphs and well-quasi-ordering[END_REF]:

Theorem 3.19. The collections of finite cographs and of finite series-parallel posets are wqo under embeddability.

This result extends to binary structures as we show below.

Lexicographical sums of binary structures.

As said, a relational structure is binary if it consists of relations which are unary or binary. If R ∶= (V, (ρ i ) i∈I ) is a binary structure and F ∶= (S x ) x∈V is a family of binary structures S x ∶= (V x , (ρ i x ) i∈I ), indexed by the elements of V such that V and the V x 's are nonempty, the lexicographic sum of F over R, denoted by ∑ x∈R S x , is the binary structure T obtained by replacing each element x ∈ V by the structure S x . More precisely, T = (Z, (τ i ) i∈I ) where Z ∶= {(x, y) ∶ x ∈ V, y ∈ V x } and for each i ∈ I, (x, y)τ i (x ′ , y ′ ) if either x ≠ x ′ and xρ i x ′ or x = x ′ and yρ i x y ′ .

Example 3.20. If R is a finite binary structure, one may use R as an operation of arity |R| acting on the class Ω µ of binary structures with signature µ, this operation R associating to each family F ∶= (S x ) x∈V the lexicographic sum of F over R. Hence, if R is a set of finite binary structures, the pair (Ω µ , R) can be viewed as an algebra. If S is a subclass of Ω µ then the subalgebra it generates, say ∑R S, consists of members of S, of lexicographical sums of members of S indexed by members of R and the iterates of these sums. Since each operation attached to R ∈ R is increasing and extensive, if R is finite, (Ω µ , R) is an algebra quasi-ordered by divisibility.

Hence, according to Theorem 3.17, we have: Lemma 3.21. If R is finite and S is wqo with respect to embeddability, then ∑R S is wqo.

3.3.3. Primality. If R ∶= (V, (ρ i ) i∈I ) is a binary structure, a subset A of V is autonomous if ρ i (x, y) = ρ i (x ′ , y) and ρ i (y, x) = ρ i (y, x ′ ) for all i ∈ I, x, x ′ ∈ A and y ∈ V ∖ A.
A binary structure is prime or indecomposable if all its autonomous subsets are trivial, that is, are either ∅, singletons, or the whole set. With this definition, structures with at most two vertices are prime. The notion of module goes back to Fraïssé [START_REF] Fraïssé | On a decomposition of relations which generalizes the sum of ordering relations Bull[END_REF] and Gallai [START_REF] Gallai | Transitiv orientbare graphen[END_REF], see also [START_REF] Fraïssé | L'intervalle en théorie des relations, ses généralisations, filtre intervallaires et clôture d'une relation[END_REF]. The notion we consider here is due to Gallai. The notion of Frassé is less demanding. A fundamental decomposition result of a binary structure into modules was obtained by Gallai [START_REF] Gallai | Transitiv orientbare graphen[END_REF] for finite binary relations (see [START_REF] Ehrenfeucht | The theory of 2-structures. A framework for decomposition and transformation of graphs[END_REF] for further extensions).

We will need the following compactness result of Ille [START_REF] Ille | Indecomposable graphs[END_REF].

Theorem 3.22. A binary structure R is prime if and only if every finite subset F of its domain extends to a finite set F ′ such that the restriction R ↾F ′ is prime.

The decompositions of a binary structure into lexicographic sums are in correspondence with the partitions of its domain into modules. Hence, if a finite binary structure is not prime, it decomposes into a sum of proper substructures indexed by some prime structure.

With Lemma 3.21, we obtain the first part of the following result.

Theorem 3.23. If a hereditary class C of finite binary structures with a finite signature contains only finitely many prime members (up to isomorphy) then:

(1) C is wqo;

(2) the cardinality of bounds of C is bounded above. Indeed, each bound must be prime and according to a result of Schmerl and Trotter [START_REF] Schmerl | Critically indecomposable partially ordered sets, graphs, tournaments and other binary relational structures[END_REF], each prime structure of size ℓ contains a prime structure of size ℓ -1 or ℓ -2. Consequently, the cardinality of bounds of C is bounded, as claimed.

Corollary 3.24. There are only countably many hereditary classes C of finite binary structures of finite signature containing only finitely many prime members (up to isomorphy).

The 4-element path is prime and every prime graph embeds a 4-element path (Sumner [START_REF] Sumner | Graphs indecomposable with respect to the X-join[END_REF] for finite graphs and Kelly [START_REF] Kelly | Comparability graphs, in Graphs and Orders[END_REF] for infinite graphs). Hence, prime cographs have at most two vertices, thus, up to isomorphism, there are only finitely many of them. The same fact holds for series-parallel graphs, hence Theorem 3.19 is a special case of (1) of Theorem 3.23.

When C is made of bichains, Theorem 3.23 is due to Albert and Atkinson [START_REF] Albert | Simple permutations and pattern restricted permutations[END_REF], the general case is in [START_REF] Oudrar | Profile and hereditary classes of relational structures[END_REF]. As we will see in Subsection 4, general results on wqo imply the boundedness of the size of bounds of several hereditary classes. The use of the Schmerl-Trotter [START_REF] Schmerl | Critically indecomposable partially ordered sets, graphs, tournaments and other binary relational structures[END_REF] result in the proof of (2) of Theorem 3.23 can be avoided. An extension of Theorem 3.23 based on a strengthening of the notion of wqo will be given in Proposition 4.12.

One could be tempted to believe that a hereditary class of finite binary structures is wqo whenever the subclass of its prime members is wqo. This is definitively false. For an example, for each integer n ≥ 2, let DF n be the double-ended fork of length n as defined in Subsection 3.7. Let C ∶=↓ {DF n ∶ n ≥ 2} be the collection of graphs which are embeddable into some DF n . The DF n 's (n ≥ 2) form an infinite antichain, hence C is not wqo. On the other hand, the prime members of C consist of all finite paths, and an independent set with two vertices, thus form a wqo class.

3.4.

Well-quasi-order and better-quasi-order. At the very begining of the study of hereditary classes, one cannot avoid facing the problem raised by Nash-Williams about better-quasi-ordering (bqo), a notion strengthening the notion of wqo that he invented and for which he proved the fundamental results. Nash-Williams (p. 700, [START_REF] St | On well-quasi-ordering infinite trees[END_REF]) asserted that "one is inclined to conjecture that most wqo sets which arise in a reasonably 'natural' manner are likely to be bqo." Problem 3. Is a hereditary class of finite structures with finite signature bqo whenever it is wqo? It is not known if the answer is positive for hereditary classes of finite graphs (it is also unsolved for the class of finite graphs equipped with the minor quasi-order while, by Robertson-Seymour's theorem, this class is wqo). It is positive for the class of bipartite permutation graphs and the corresponding classes of bichains and posets mentioned in Theorem 3.45 (the proof is nontrivial, see [START_REF] Pouzet | Hereditary classes of ordered sets coverable by two chains[END_REF]). It is also positive for the ages of almost multichainable relational structures studied in Section 5 (see Theorem 5.34). However, the answer is negative if the signature is infinite and the arity is unbounded. There are wqo ages that are not bqo. Examples are based on a construction given in [START_REF] Pouzet | Sandwiches of ages[END_REF]. We recall it in Subsection 3.4.1.

We give the minimal information needed to understand bqo. We recall that if P is a poset, the Higman ordering defined in Subsection 1.4.3 on finite sequences can be extended to ordinal sequences. If f ∶ α → P and g ∶ β → P are two ordinal sequences, set f ≤ g if there exists some injective order preserving map h ∶ α → β such that f (γ) ≤ g(γ) for all γ < α. This gives a quasi-order. Given an ordinal α, P is α-wellquasi-ordered (α-wqo for short) if the set P α of ordinal sequences f ∶ α → P is well-founded. Say that P is better-quasi-ordered (bqo) if it is ω 1 -wqo for the first uncountable ordinal ω 1 . This is the shortest possible definition of bqo. It is not too difficult to prove that if α is an indecomposable ordinal then P is α-wqo if and only if the collection P <α of β-sequences f ∶ β → P , with β < α, is wqo (use Higman's theorem on words, see Theorem 3.18). For an example, P is ω-wqo if and only if it is wqo. Since ω 1 is an indecomposable ordinal, a poset P is bqo if and only if the set P <ω 1 is wqo. More is true, Nash-Williams [START_REF] St | On well-quasi-ordering infinite trees[END_REF] proved that if P is bqo then the collection of all ordinal seqences with values in P is bqo. For more, see the paper by Marcone [START_REF] Marcone | Foundations of bqo theory[END_REF].

A more intuitive definition of bqo presented by Laver [START_REF] Laver | On Fraïssé's order type conjecture[END_REF] is that the transfinite iterates I [β] (P ) of the set of initial segments of P , where β runs throught ordinal numbers, are all wqo. 3 None of these definitions are easy to work with; the appropriate tools for the notion of bqo are based on the notion of barrier invented by Nash-Williams [START_REF] St | On well-quasi-ordering infinite trees[END_REF].

A basic example of a wqo poset P such that I(P ) is not wqo was discovered by Rado. This is the set

Rad ∶= {(n, m) ∈ [N] 2 ∶ n < m} ordered by (n, m) ≤ (n ′ , m ′ ) if either n = n ′ and n ≤ m ′ or m < n ′ .
Marcone [START_REF] Marcone | Foundations of bqo theory[END_REF] proved that:

Theorem 3.25. For each countable indecomposable ordinal α there is some poset P α made of finite subsets of ω which is not α-wqo but is β-wqo for every β < α.

For an example, Rad is not ω 2 -wqo, but is β-wqo for every β < ω 2 . From Marcone's result it follows that for each countable indecomposable ordinal α there is some poset Q made of finite subset of ω such that all I [β] (Q) are wqo for β < α and

I [α] (Q) is not wqo.
The notion of bqo was invented to prove that some classes of countable structures are wqo (see Section 8 below), so one may ask why it is important to know whether wqo ages which are made of finite structures are in fact bqo. The reason is twofold. First, in several instances, the complexity of an infinite hereditary class C of finite structures can be better represented by a single infinite relational structure R (this requiring that C is an age and C = Age(R)). This is one of the originalities of the approach of combinatorial problems by the theory of relations. Next, properties of families of ages can be reflected in properties of families of countable structures. For example, countable antichain of ages yield trivially countable antichains of countable structures (the same property holds with chains, see Theorem V-2.3 in [START_REF] Pouzet | Chaînes de théories universelles[END_REF] and also [START_REF] Pouzet | Sandwiches of ages[END_REF]). Now, if an age embeds the Rado poset, it will contain an infinite antichain of ages. Up to now, the known examples of ages embedding this poset and which are wqo are made of relational structures with an unbounded signature. Such an example is given below.

3.4.1.

A construction of wqo ages which are not bqo. The construction below is borrowed from [START_REF] Pouzet | Sandwiches of ages[END_REF], p.301.

Let V be an infinite set and

F ⊆ [V ] <ω ∖ {∅}. Let M F ∶= (V, (K F ) F ∈F ) be the relational structure, where K F is the |F |-ary relation on V defined by K F (x 1 , . . . , x |F | ) = + if and only if {x 1 , . . . , x |F | } = F . Let U F ∶= (V, (U F ) F ∈F ) where U F (x 1 , . . . , x |F | ) = -everywhere.
Let F <ω (resp., F ∪ ) be the collection of finite (resp., arbitrary) unions of members of F.

Given A ⊆ V , set ○ A∶= ⋃{F ∈ F ∶ F ⊆ A}.
Lemma 3.26. The age Age(M F ) of M F ordered by embeddability embeds the set F <ω ordered by set inclusion. It is the image of the direct product F <ω × ω by an order preserving map provided that for every F ∈ F <ω and every integer n ∈ ω there is some

H ∈ [V ∖ F ] n such that ○ (F ∪ H)= F .
Proof. The first assertion in the lemma is obvious. For the second assertion, let

(F, n) ∈ F <ω × ω. Observe that if H, H ′ ∈ [V ∖F ] n such that ○ F ∪ H= F and ○ F ∪ H ′ = F , the restrictions M F ↾ F ∪ H and M F ↾ F ∪ H ′ are isomorphic. Hence we may set f (F, n) = M F ↾ F ∪ H
and by this way define an an order preserving map from F <ω × ω in Age(M F ). This map is trivially surjective since for every subset

X ∈ [V ] <ω , the restriction M F ↾ X is of the form f ( ○ X, n) where n = |X∖ ○ X |.
Corollary 3.27. The age Age(M F ) is wqo if and only if F contains no infinite antichain.

Proof. Since F ordered under set inclusion is well-founded, it is wqo provided it contains no infinite antichain. By Higman's Theorem, F <ω is wqo. Hence, the direct product F <ω × ω is wqo. By Lemma 3.26, Age(M F ) is the image of the direct product F <ω × ω. Thus, Age(M F ) is wqo. Conversely, suppose that Age(M F ) is wqo. Since it embeds F <ω , this set is wqo. Hence F is wqo.

Due to Lemma 3.26, if F is wqo but not bqo, the age of Age(M F ) is wqo and not bqo.

• Example of an age A which is wqo and such that I(A) is not wqo.

Let F ∶= {{0, . . . , j} ∖ {i} ∶ 0 ≤ i < j < ω}. Ordered by inclusion, F is isomorphic to Rad. Since Rad is wqo, the age A ∶= Age(M F ) is wqo by Corollary 3.27. But, since Rad embeds in A, I(Rad) embeds in I(A), hence this set is not wqo.

• Example of an age A such that I(A) is wqo but I(I(A)) is not.

Let P be a poset embeddable into [ω] <ω such that I(P ) is wqo but I(I(P )) is not. Set V ∶= P ∪ A, where A is an infinite set disjoint from P . Let F ∶= {↓ x ∶ x ∈ P }. Let M F be as above and A ∶= Age(M F ). In terms of the theory of bqo, F, that is P , is ω 2 -wqo but not ω 3 -wqo. From the extension of Higman's theorem to bqo's, it follows that F <ω is ω 2 -wqo and hence

F <ω × ω is ω 2 -wqo. Since A is a surjective image of F <ω × ω, it is ω 2 -wqo, that is I(A)
is wqo. Since P embeds in A and P is not ω 3 -wqo, A is not, hence I(I(A)) is not wqo.

• A more complicated example follows from Theorem 3.25: For every indecomposable ordinal α there is an age A such that all I [β] (A) are wqo for β < α and I [α] (A) is not wqo. However, our examples have unbounded signature (and an infinite kernel). In order to be more specific about this problem, we gather in the theorem below obtained with Sobrani [START_REF] Pouzet | Ordinal invariant of an age[END_REF] various properties of the set of ages included in a given hereditary class.

Theorem 3.28. Let us consider the following properties of a countable hereditary class C of finite structures:

(i) C is bqo; (ii) Id(C) is wqo; (iii) C is wqo; (iv) Id(C) is a countable Priestley space; (v) Id(C) is topologically scattered; (vi) Id(C) is order-scattered; (vii) every chain of Id(C) is at most countable; (viii) Id(C) is well-founded; (ix) [ω] <ω does not embed into C; (x) ℘(ω) does not embed into Id(C).
Then:

(i) ⇒ (ii) ⇒ (iii) ⇒ (iv) ⇒ (v) ⇒ (vi) ⇔ (vii) ⇔ (viii) ⇒ (ix) ⇔ (x).
• There are ages C of structures of unbounded signature such that

(iii) ⇏ (ii) ⇏ (i).
• There are hereditary classes C of finite structures with a finite signature such that Id(C) is well-founded and has cardinality 2 ℵ 0 proving that (viii) ⇏ (v).

Proof. Positive results. The really nontrivial implication is (vii) ⇒ (viii) ( [START_REF] Pouzet | Chaînes de théories universelles[END_REF], Section II-3.1) and also ([150], Théorème V-4, p. 76). Implications (i) ⇒ (ii) ⇒ (iii) are elementary facts about wqo. (iii) ⇒ (iv). Since C is wqo, it is level-finite, we may apply Proposition 2.13. Hence, Id(C) is a countable Priestley space. (iv) ⇒ (v). Since Id(C) is countable and compact, it is topologically scattered (see Subsection 1.2). (v) ⇒ (vi). Suppose by way of contradiction that there is some embedding f from the chain Q of rational numbers into Id(C). Since Id(C) is closed under unions of chains, for every r ∈ R the set f (r) ∶= ⋃{f (r ′ ) ∶ r ′ < r} belongs to Id(C). This allows to define an embedding f ∶ R → Id(C). Since the image of this map has no isolated point, Id(C) is not topologically scattered. (vi) ⇔ (vii). If Id(C) is order scattered, an uncountable chain would contain a chain with order type ω 1 or its dual ω * 1 . But, this will lead to a sequence of ℵ 1 distinct elements of C. Since C is countable, this is impossible. (vii) ⇔ (viii). That (vii) ⇒ (viii) is treated in Section II-3, Proposition II.3.2 on page 143 of [START_REF] Pouzet | Chaînes de théories universelles[END_REF]. It makes use of a special property of ideals that we will consider in Subsection 3.6.1 (cf. Theorem 3.35), namely, the fact that for every ideal P ∈ Id(C) and x ∈ P , the set g(x, P ) of Q ∈ Id(P ) such that {x} ∪ Q generates P is finite (that is no proper ideal of P contains {x} ∪ Q). To see the converse, note that (viii) ⇒ (vi) is trivial and we have the implication (vi) ⇒ (vii) above. Finally, (viii) ⇒ (x) is obvious since ℘(ω) is not well-founded. The equivalence (ix) ⇔ (x) is Theorem 1.1, a general fact about posets.

Negative results. The examples given in Subsection 3.4.1 prove our first assertion. For the second, by no means trivial, an example of a hereditary classe C of finite structures with a finite signature such that Id(C) is well-founded and has cardinality 2 ℵ 0 is given in Proposition 7.4.

We note that Theorem 3.40 shows that the implication (v) ⇒ (viii) remains true when the embeddability order is replaced by set containment.

Since level-finite hereditary classes are countable, they embed in [ω] <ω (to each τ belonging to a hereditary class C associate the finite set Here are two subproblems, formulated in 1978, of this conjecture; see [START_REF] Pouzet | Condition de chaîne en théorie des relations[END_REF], I.4, p. 67.

↓ τ = {τ ′ ∈ C ∶ τ ′ ≤ τ }).
Problem 5. Determine whether an age is wqo whenever (1) the collection of its subages is well-founded with respect to set inclusion, or (2) the collection of its subages is countable. Implication (ii) ⇒ (v) of Theorem 3.28 asserts that condition (2) implies condition [START_REF] Abdi | An example of Tateno disproving conjectures of Bonato-Tardif, Thomassé and Tyomkyn[END_REF]. But, the solutions to these problems elude us.

3.6. Topological scatteredness versus well-quasi-ordering: an illustration of the notion of kernel. In this subsection, we show that the topological scatteredness of the space of subages of a level-finite hereditary class of Ω µ implies the well-foundedness of that space ordered by set inclusion (Theorem 3.40). Proofs are based on the notions of kernel and generation.

3.6.1. Kernel. Let R be a relational structure with domain V . For X ⊆ V , set R -X ∶= R ↾V ∖X ; if X ∶= {x}, resp., X ∶= {x, y} set R -x ∶= R -X , resp., R -x,-y ∶= R -X . If S ∈ Age(R), the kernel of S in R is the set ker(S, R) of x ∈ V such that S does not embed in R -x . The kernel of R, ker(R), is the set union of the kernels ker(S, R) for S ∈ Age(R), that is, ker(R) ∶= {x ∈ V ∶ Age(R -x ) / = Age(R)}. Let a, b ∈ V (R). Set a ≤ R b if Age(R -a ) ⊆ Age(R -b
). This defines a quasi-order on V (R). It plays a crucial role in the study of the kernel. We mention the following fact (see [START_REF] Pouzet | Relation minimale pour son âge[END_REF], III.1.5). Other properties will be recalled in Subsection 5.4. Lemma 3.30. Let R be a relational structure with domain V and a, b ∈ V .

(

) If b / ∈ ker(R) that is Age(R -b ) = Age(R) then Age(R -a,-b ) = Age(R -a ); (2) If b ∈ ker(R) then the initial segment {a ∈ V ∶ a ≤ R b} is finite. Lemma 3.31. If the kernel of R is infinite then: (1) the set {Age(R -a ) ∶ a ∈ ker(R)} is infinite; 1
(2) there are arbitrarily long finite chains of infinite subages of Age(R).

Proof. (1) is an immediate consequence of (2) of Lemma 3.30.

(2). Define a sequence (a n , A n ) n∈N . Pick a 0 ∈ ker(R) and A 0 finite such that R ↾A 0 / ≤ R -a 0 . Suppose the sequence is defined for all n ′ < n. Pick a n ∈ ker(R) ∖ ⋃ n ′ <n A n ′ and pick A n finite such that R ↾An / ≤ R -an . Let k ∈ N. We have: (3.1) Age(R -a 0 ,...,-a k ) ⊆ Age(R -a 1 ,...,-a k ) ⊆ ⋅ ⋅ ⋅ ⊆ Age(R -a i ,...,-a k ) ⊆ ⋅ ⋅ ⋅ ⊆ Age(R -a k ). Since R ↾A i ∈ Age(R -a i+1 ,...,-a k ) ∖ Age(R -a i ,.
..,-a k ) for i = 0, . . . , k -1, this sequence of ages is strictly increasing with respect to set inclusion.

The kernel is an invariant of the age of R in the following sense: If R and R ′ have the same age, then there is an isomorphism f from R ↾ker(R) onto R ′ ↾ker(R ′ ) such that (a) every restriction of f to every finite subset F of ker(R) extends to every finite superset F of F to an embedding of R ↾F in R ′ ; (b) the same property holds for f -1 .

From (1) of Lemma 3.30, for each finite subset F of V disjoint from the kernel of R, the restriction R ↾V ∖F has the same age as R.

Here are two other lemmas that can be found in [START_REF] Pouzet | Relation minimale pour son âge[END_REF], III. 1.3 and [152], III.1.6, p. 323. Lemma 3.32. An element a ∈ V (R) belongs to ker(R) if and only if there is some finite subset A of V (R) containing all the images of a by the local automorphisms defined on A.

Lemma 3.33. The kernel of R is the least subset with respect to set inclusion among the subsets K of V (R) such that for every nonempty subset H of V (R) ∖ K and every finite subset A of K there are infinitely many local isomorphisms defined on H, extendible via the identity on A and with pairwise disjoint images.

An age A is inexhaustible, or has the disjoint embedding property, if two arbitrary members of A can be embedded into a third member in such a way that the images of their domains are disjoint. Due to the fact above, the kernel of a relational structure R is empty if and only if Age(R) is inexhaustible. We say that an age A which is not inexhaustible is exhaustible. It is almost inexhaustible if the kernel of some R with age Age(R) = A is finite.

Almost inexhaustible and inexhaustible structures are linked by means of reductions, which are defined as follows. Given a relational structure R ∶= (V, (ρ i ) i∈I ) and K ⊆ V a reduction of R over K is a relational structure M ∶= (V ∖ K, (τ j ) j∈J ) such that the local automorphisms of M are precisely the restrictions of local isomorphisms of R fixing K pointwise (this is not to be confused with the notion of reduct). Lemma 3.34. Let R be a relational structure, ker(R) be its kernel, r ∶= | ker(R)|, and M be a reduction of R over ker(R). If ker(R) is finite then:

(1) ker(M ) is empty.

(2) H(Age(R)) = H(Age(M )) + p, for some integer p, if and only if one of these heights exists.

(

) φ R (n) ≤ 2 r φ M (n) and φ M (n) ≤ aφ R (n + k) for some a ∈ R * + , k ∈ N, 3
and all n ∈ N. We note that (1) is special case of Theorem 20 of [START_REF] Gibson | Relational structures having finitely many full-cardinality restrictions[END_REF], (2) is a special case of Theorem 4.6 of [START_REF] Pouzet | Sandwiches of ages[END_REF], and (3) can be proved in the same way as Theorem 21 of [START_REF] Gibson | Relational structures having finitely many full-cardinality restrictions[END_REF], using the fact that the profile does not decrease.

The notion of inexhaustibility was introduced by Fraïssé in the 1960's. The notion of kernel appears first in the special case of almost chainable structures studied by Fraïssé. He proved the existence of a smallest subset K (smallest with respect to set inclusion) for which an almost chainable structure was K-chainable, see Subsection 5.1 ). The general notion was introduced in [START_REF] Pouzet | Sur la théorie des relations[END_REF] and studied in several papers [START_REF] Pouzet | Relation minimale pour son âge[END_REF], [START_REF] Pouzet | Application de la notion de relation presque-enchaînable au dénombrement des restrictions finies d'une relation[END_REF], and Lemma IV-3.1, p. 37, [START_REF] Pouzet | Relation impartible[END_REF], first for structures with finite signature. The general case was considered in [START_REF] Pouzet | Sandwiches of ages[END_REF].

3.6.2. Generation of an age. Let A ′ ⊆ A be two ages and S ∈ A ∖ A ′ . We say that A is generated by A ′ and S if there is no proper subage of A containing A ′ ∪ {S}. When these conditions are fulfilled then [START_REF] Pouzet | Sandwiches of ages[END_REF], Proposition 3.9). But, in principle, without some conditions, the fact that A ′ ⊆ A does not imply the existence of R and R ′ such that R ′ is a restriction of R, Age(R ′ ) = A ′ and Age(R) = A. Such a condition is that A is level-finite. So, in this subsection we suppose that the hereditary classes under consideration are level-finite.

(3.2) ker(R) = ker(R ′ ) ∪ ker(S, R) for every R ′ , R such that R ′ is a restriction of R, Age(R ′ ) = A ′ and Age(R) = A (see
Let A be an age and S ∈ A. We denote by g(S, A) the set of ages A ′ which together with S generate A.

We recall Proposition V-2.3 of [START_REF] Pouzet | Condition de chaîne en théorie des relations[END_REF], and Corollary 3.10 of [START_REF] Pouzet | Sandwiches of ages[END_REF]. In [START_REF] Pouzet | Sandwiches of ages[END_REF], Theorem 3.16 p. 314, it was shown that the interval [A ′ , A] contains an infinite chain (and in fact a chain of order type ω) whenever it is infinite. More precisely: Theorem 3.37. Two ages A ′ and A be given with A ′ ⊂ A be given, if the set [A ′ , A] of ages B such that A ′ ⊆ B ⊆ A is infinite, then this set contains a chain of ages with order type ω + 1.

We will use this result for the proof of Proposition 5.49 below. We will use an other consequence (part (a) is Proposition 4.7, p. 325 of [START_REF] Pouzet | The profile of relations[END_REF]). Proof. (a). Suppose that H(A) = α + 1. There is some A ′ ∈ Id(A) such that A ′ = α. Then the interval [A ′ , A] has two elements. According to Theorem 3.36, there is some S ∈ A such that A ′ and S generate A. If R ′ , R are such that Age(R ′ ) = A ′ and Age(R) = A then, from Equation 3.2, ker(R) = ker(R ′ ) ∪ ker(S, R). Hence ker(S, R) / = ∅. This proves that A is not inexhaustible. (b). This is an immediate consequence of (a).

3.6.3. Height of an age and Cantor-Bendixson rank. A link between the height of an age and its Cantor-Bendixson rank (when defined) is given by the following proposition. The first part is Proposition V-3.2 in [START_REF] Pouzet | Condition de chaîne en théorie des relations[END_REF] (with a small improvement). The second part is in [START_REF] Pouzet | Ordinal invariant of an age[END_REF].

Proposition 3.39. Let C be a hereditary class of finite structures which is level-finite, Id(C) be the set of ages included in C, equipped with the product topology, α be an ordinal and A ∈ Id (α) (C).

(1) A is above some minimal element of Id (3) If A has finitely many bounds and Id(A) is wqo then A has a Cantor-Bendixson rank in Id(C) and if β is its rank, then

(3.3) ω.β ≤ H(A) < ω ⋅ (β + 1).
Proof. [START_REF] Abdi | An example of Tateno disproving conjectures of Bonato-Tardif, Thomassé and Tyomkyn[END_REF]. Since C is level-finite, Id(C) is compact for the product topology (Proposition 2.13). Being closed, Id (α) (C) is compact too, hence closed under the intersection of every chain of ages. Zorn's lemma ensures that every element of Id (α) (C) contains some element, minimal for set inclusion. If A is a minimal element of Id (α) (C) then no element of Id(A) ∖ {A} is in Id (α) (C), thus all are in Id (<α) (C). If A has finitely many bounds in C then there is a clopen set reduced to Id(A), hence A has Cantor-Bendixson rank α. If, in general, we cannot ensure that A has Cantor-Bendixson rank, then since every A ′ ⊂ A has some rank α ′ < α, the truth of (2) proved below ensures that every Id(A ′ ) is well-founded, hence Id(A) is well-founded, and

H(A ′ ) ≤ ω.α ′ + p ′ < ω.α. Hence, H(A) ≤ ω.α.
The proofs of (2) and (3) are by induction on α.

(2). If α = 0 then, as it is easy to see, A is the age of a finite structure, hence Id(A) is finite and thus, A is well-founded, H(A) < ω and our assertion holds. Let α > 0. Suppose that our assertion holds for every α ′ < α. Our aim is to prove that there exists some integer n such that Id(A ′ ) is well-founded and

H(A ′ ) ≤ ω.α + n for all A ′ ∈ Id(A). Since rank(A, Id(C)) = α, there is a clopen set U ⊆ Id(C) containing A such that rank(A ′ , Id(C)) < α for every A ′ ∈ U ∖ {A}. Such a clopen set U is of the form [F, C ∖ H) for some finite subsets F and H of C. Since A ⊇ F and F is finite, there is some S ∈ A which dominates F . Let A ′ ⊆ A. If S ∈ A ′ / = A then α ′ ∶= rank(A ′ , Id(C)) < α. Via the inductive hypothesis, Id(A ′ ) is well-founded and H(A ′ ) < ω.(α ′ + 1) ≤ ω.α. If S / ∈ A ′ , then Id(A ′
) must be well-founded. Otherwise, since by Theorem 3.35, g(S, A) is finite, Id(A ′ ) contains some A ′′ such that Id(A ′′ ) is not well-founded and, with S, does not generate A. So, there is an age B such that S ∪ A ′′ ⊆ B ⊂ A. Since rank(B, Id(C)) < α, the induction hypothesis ensures that Id(B) is well-founded. Since this is not the case, we have a contradiction. To achieve our aim, suppose by contradiction that, for each integer n, there is some A n ∈ Id(A) not containing S such that H(A n ) ≥ ω.α + n. Since by Theorem 3.35, g(S, A) is finite, there is some n and some age B such that {S} ∪ A n ⊆ B ⊂ A. Since rank(B, Id(C)) < α, induction hypothesis ensures that H(B) < ω.α.

Since A n ⊆ B, H(A n ) ≤ H(B) < ω.α, which is impossible.
(3). Since A is wqo, Id(A) is topologically scattered. Hence, every member of Id(A) has a rank in Id(A). Since A has finitely many bounds in C, it has the same rank in Id(C) as in Id(A). If α = 0, whatever A is, H(A) ≥ 0, hence the inequalities hold. Suppose α > 0. By definition of the rank of A there is a clopen set

U ⊆ Id(C) containing A such that α = sup{rank(A ′ , Id(C)) + 1 ∶ A ′ ∈ U ∖ {A}}.
We consider two cases. Case 1. α is a successor ordinal, that is, α = α ′ + 1. In this case U contains an infinite sequence (A n ) n<ω with rank(A n , Id(C)) ≥ α ′ for each n < ω. Since Id(A) is wqo, we may extract a sequence (A φ(n) ) n<ω which is increasing with respect to set inclusion. Via the induction hypothesis,

H(A φ(0) ) ≥ ω.α ′ . Hence H(A) ≥ ω.α ′ + 1 + ⋅ ⋅ ⋅ + 1 + ⋅ ⋅ ⋅ = ω.α ′ + ω = ω.(α ′ + 1) = ω.α as required. Case 2. α is a limit ordinal. In this case α = sup{α n ∶ n < ω } with each α n < α. For each n < ω, we may select A n ⊂ A with rank(A n , Id(C)) ≥ α n . Via the induction hypothesis, H(A n ) ≥ ω.α n . Since H(A) ≥ H(A n ) the inequality H(A) ≥ ω.α follows.
The following problem is open, see V-5.2, p. 77 in [START_REF] Pouzet | Condition de chaîne en théorie des relations[END_REF]. Problem 6. Let A ′ ⊆ A be two ages. If A ′ has a rank in Id(A), does every age A ′′ included in A ′ has a rank in Id(A) and, if this is the case, does the inequality rank(A ′′ , Id(A)) ≤ rank(A ′ , Id(A)) hold?

The first inequality in Equation 3.3 cannot hold without some condition on A. This is clear if instead of an age we consider a level-finite poset P . If P is the Rado poset, then rank(P, Id(P )) = 2 and h(P, Id(P )) = ω + 1. But if P ∶= [ω] 2 (the subset of the direct product ω × ω ordered componentwise), rank(P, Id(P )) = 2 and h(P, Id(P )) = ω ⋅ 2. The first example given in 3.4.1 yields a concrete example of an age falsifying the stated inequality but the kernel is infinite and the signature is unbounded. In Section 5.4, it will be shown that ages of height less than ω 2 have only finitely many bounds (Theorem 5.34 and Theorem 5.42) and the collection of their subages is wqo (Theorem 5.34). With that fact and the above result we obtain the following result. (This was announced in [START_REF] Pouzet | Condition de chaîne en théorie des relations[END_REF] and a proof given in Corollaire p.6 in "Caractérisation combinatoire et topologique des âges les plus simples" [START_REF] Pouzet | Sur la théorie des relations[END_REF] but not published.) Theorem 3.41. Let C be a level-finite hereditary class of finite structures and n be an integer. A is the age of an almost multichainable structure, hence it has finitely many bounds and the collection of subages is wqo. According to (3) of Proposition 3.39, H(A) ≥ ω ⋅ n. Conversely, if Id(A) is well-founded and ω ⋅ n ≤ H(A) < ω ⋅ (n + 1), then due to Theorem 5.34 and 5.42, A has finitely many bounds and the collection of subages is wqo. Thus, in virtue of (3), A has rank n. Let us prove that (2) holds. Let U be a clopen set intersecting Id (n) (C). Take a minimal element A w.r.t. set inclusion. Apply (1) of Proposition 3.39. Then, H(A) < ω ⋅ (n + 1). As said, A has finitely many bounds, hence by (3) of Proposition 3.39, rank(A,

Id(C)) = n, that is A is isolated in Id (n) (C).
3.6.4. Vietoris topology. Sobrani and I observed that the Vietoris topology seems to be appropriate to study ideals of a poset when the collection of ideals is well-founded and not necessarily well-quasi-ordered. We give some hints. Details are in [START_REF] Pouzet | Ordinal invariant of an age[END_REF], a paper to appear soon.

Let P be a set and F, H

⊆ P with F ⊆ H. Set [F, H] ∶= {X ∈ ℘(P ) ∶ F ⊆ X ⊆ H}. The collection {[F, H] ∶ F ∈ [P ]
<ω , H ⊆ P } is the basis of a topology on ℘(P ), the Vietoris topology. Since the collection {[F, P ∖ H] ∶ F, H ∈ [P ] <ω } is a basis of the product topology, the Vietoris topology is finer than the product topology.

If P is wqo, there is no need to consider the Vietoris topology induced on the set Id(P ) of ideals of P . Indeed: Lemma 3.42. Let P be a poset. Then the following properties are equivalent:

(i) P is wqo. (ii) The Vietoris topology and the product topology induced on I(P ) coincide. (iii) (a) The Vietoris topology and the product topology induced on Id(P ) coincide.

(b) Id(P ) is compact w.r.t. the product topology.

We may apply the Cantor-Bendixson reduction process to Id(P ) equipped with the Vietoris and product topologies induced by ℘(P ). It is immediate to observe that an element Q ∈ Id(P ) is isolated for the Vietoris topology if and only if Q =↓ a for some a ∈ P . This generalizes as follows: to each nonempty up-directed poset Q associate, whenever possible, an ordinal, rank V (Q), the Vietoris rank, by the following rules:

• rank V (Q) ∶= 0 if Q has a largest element; • assuming that all Q ′ such that rank V (Q ′ ) < α are defined, we set rank V (Q) ∶= α if rank V (Q)
is not strictly less than α and there is some element

x ∈ Q such that rank V (Q ′ ) < α for each Q ′ ∈ Id x (Q) ∖ {Q}, where Id x (Q) ∶= {Q ′ ∶ x ∈ Q ′ ∈ Id(Q)}.
For example, if Q is up-directed, rank V (Q) = 1 if and only if Q has no largest element and there is some

x ∈ Q such that every proper ideal of Q containing x has a largest element.

Lemma 3.43. Let P be a poset and Q ∈ Id(P ). Then rank V (Q) is the topological rank of Q in Id(P ) equipped with the Vietoris topology.

Since the Vietoris topology is finer than the product topology, Id(P ) is scattered for the Vietoris topology whenever it is scattered for the product topology. The converse does not hold (think about the binary tree). For hereditary classes, we have: Theorem 3.44. For every level-finite hereditary class C of finite structures, the set of ages Id(C) is scattered for the Vietoris topology if and only if it is well-founded with respect to set inclusion.

3.7.

The case of ages of posets coverable by two chains. Recently, Imed Zaguia and I [START_REF] Pouzet | Hereditary classes of ordered sets coverable by two chains[END_REF] realized that the four types of ages described in the introduction of this section are equal for ages consisting of finite posets coverable by two chains. We also show that the answer to Conjecture 1 is positive. The full result reads as follows.

Theorem 3.45. Let B be a bichain, P ∶= o(B) and G ∶= Inc(P ). If P is coverable by two chains and C is the age of one of these structures and if Id(C) is the collection of ages included into C then the following propositions are equivalent.

(i) C is bqo;

(ii) C is wqo; (iii) Id(C) is countable; (iv) Id(C) is well-founded; (v) ℘(N) does not embed in Id(C); (vi) [ω] <ω does not embed in C.
The equivalence of (ii) and (vi) is in Corollary 50 of [START_REF] Pouzet | Hereditary classes of ordered sets coverable by two chains[END_REF]. The equivalence between (i) and (ii) is a consequence of Theorem 10 of [START_REF] Pouzet | Hereditary classes of ordered sets coverable by two chains[END_REF]. Theorem 3.28 asserts that the implications (i) ⇒ (ii) ⇒ (iii) ⇒ (iv) ⇒ (v) ⇒ (vi) hold. We will give a hint of the implication ¬(ii) ⇒ ¬(vi), which follows from properties of double-ended forks.

A double-ended fork DF n of length n, for n ≥ 1, is the graph obtained from a path P n on n vertices by adding two vertices adjacent to each end vertex of P n , these two vertices being nonadjacent.

...

FIGURE 1. Double-ended forks

Theorem 3.46. Let C be an age made of finite bipartite permutation graphs. Then C is not wqo if and only if it contains the age of a direct sum ⊕ i∈I DF i of double-ended forks of arbitrarily large length for some infinite subset I of N.

With this in hand, the proof of the implication ¬(ii) ⇒ ¬(vi) goes as follows. Suppose that C = Age(G) and (ii) does not hold. Apply Theorem 3.46. Since C contains an infinite antichain, there is a infinite subset I of N such that Age(⊕ n∈I DF n ) ⊆ C. For every finite subset F of I, set φ(F ) ∶= ⊕ n∈F DF n . This leads to an embedding of [I] <ω , the set of finite subsets of I, ordered by set inclusion, in Id(C), hence an embedding of [ω] <ω in Id(C), since, as posets, [I] <ω and [ω] <ω are isomorphic. Hence (v) does not hold. Now, if Age(B) or Age(P ) are not wqo, note that Age(G) is not wqo. To the infinite antichain of DF n with n ∈ I as above corresponds an antichain in Age(B) and in Age(P ) and it yields and embedding of [ω] <ω in Age(B) and in Age(P ). Hence ¬(ii) ⇒ ¬(vi) holds for C = Age(B) and for Age(P ) as well.

Theorem 3.46 above is Theorem 14 of [START_REF] Pouzet | Hereditary classes of ordered sets coverable by two chains[END_REF]. It extends previous results of Korpelainen and Lozin, Theorem 26 in [START_REF] Korpelainen | Bipartite induced subgraphs and well-quasi-ordering[END_REF], and of Lozin and Mayhill, Theorem 7 in [START_REF] Lozin | Canonical antichains of unit interval and bipartite permutation graphs[END_REF].

Problem 8. Extend Theorem 3.45 to arbitrary bichains.

3.7.1.

A parallel development: ideals in join-semilattices. We do not know how to characterize Priestley spaces made of ages of a hereditary class. Ages have much in common with ideals of a join-semilattice. In terms of orders, these ideals form an algebraic lattice. With Sobrani [START_REF] Pouzet | Ordinal invariant of an age[END_REF] and Chakir [START_REF] Chakir | Infinite independent sets in distributive lattices[END_REF], [START_REF] Chakir | Chains conditions in algebraic lattices[END_REF] we studied the length of chains in algebraic lattices. Replacing in Theorem 3.28 the poset Ω µ by the lattice I <ω (Q) of a well-founded poset Q and replacing the age C by the join-semilattice P of I <ω (Q) generated by some subposet of Q, we proved in that case that the properties in Theorem 3.28 are equivalent (see Theorem 4.16 and Corollary 4.17 of [START_REF] Chakir | Infinite independent sets in distributive lattices[END_REF]). We deduced the following result (Corollary 1.8): Lemma 3.47. A join-subsemilattice of [ω] <ω contains either [ω] <ω as a join-subsemilattice or is wqo. Some examples of ages fit in this frame. For example, let P be made of finite direct sums of finite chains; as a poset it is isomorphic to I <ω (ω × ω). Then Id(P ) is isomorphic to I(ω × ω). Hence H(P ) ∶= h(P,

Id(P )) = h(ω × ω, I(ω × ω)) = o(ω × ω) = ω 2 , while o(Id(P )) = ω ω+2 + 1.
Other examples were given in 3.4.1.

The exact relationship between the ordinal lengths of a well-quasi-ordered subset A of [ω] <ω and the poset I <ω (A) of finitely generated initial segments of A can be obtained. Write o(A) = ω ⋅ ξ + p where p is the cardinality of the largest finite final segment Ǎ of A. Then o(I <ω (A)) = ω ξ ⋅ q where q is the cardinality of F( Ǎ), the set of final segments of Ǎ (see [START_REF] Pouzet | Ordinal invariant of an age[END_REF]).

LABELLING AND WELL-QUASI-ORDERING

In this section, we discuss the preservation of well-quasi-order by labelling. There are several notions for which the eventual equivalence is not known. When they hold they have some strong consequences, notably on the finiteness of bounds of hereditary classes, as will be illustrated in Section 5.

4.1.

Extensions of well-quasi-ordering to labelled classes. The concatenation of two relational structures R and S on the same domain is the relational structure, denoted by R ⋅ S, made of the relations of R followed by the relations of S. Let P be a poset. A labelling of a relational structure R by P is a map f from V (R), the domain of R, into P . Let C be a class of relational structures with a fixed signature µ. Denote by C ⋅ P the class of relational structures R ∈ C labelled by P , that is, the class of pairs (R, f ) with f ∶ V (R) → P . We quasi-order C ⋅ P by dominance:

(4.1) (R, f ) ≤ (R ′ , f ′ ) if there is an embedding h ∶ R → R ′ such that f (x) ≤ f ′ (h(x)) for all x ∈ V (R).
Examples 4.1. Suppose that P = {0, 1} with 0 incomparable to 1. Then C ⋅ P with this quasi-order is isomorphic to the collection C [START_REF] Abdi | An example of Tateno disproving conjectures of Bonato-Tardif, Thomassé and Tyomkyn[END_REF] made of concatenats R ⋅ U where R ∈ C and U is a unary relation on V (R), this collection being quasi-ordered by embeddability. More generally, let m be an integer and P be an antichain of size 2 m then C ⋅ P with the quasi-order above is isomorphic to the collection C [m] made of concatenats R ⋅ U 1 ⋯U m where R ∈ C and U 1 , . . . , U m are m unary relation on V (R), this collection being quasi-ordered by embeddability. Indeed, let us identify the 2 m -element antichain P with ℘({1, . . . , m}), the power set of {1, . . . , m}, ordered by the equality relation. To

f ∶ V → ℘({1, . . . , m}) associate U (f ) ∶= (U 1 , . . . , U m ) where U i ∶= {x ∈ V (R) ∶ i ∈ f (x)}. Clearly (R, f ) ≤ (R ′ , f ′ ) if and only R ⋅ U (f ) ≤ R ′ ⋅ U (f ′ ).
More interestingly, order ℘({1, . . . , m}) with the inclusion order, in this case

R ⋅ U 1 ⋯U m ≤ R ′ ⋅ U ′ 1 ⋯U ′ m means that there is some embedding h ∶ R → R ′ such that x ∈ U i implies h(x) ∈ U ′ i for every x ∈ V (R) and i ∶= 1, . . . , m.
Let R be a relational structure and m be a nonnegative integer. A labelling of R with m constants is the structure R with m distinguished elements a 1 , . . . , a m of V (R) added. An embedding of (R,

a 1 , . . . a m ) into (R ′ , a ′ 1 , . . . a ′ m ) is an embedding f of R into R ′ such that f (a i ) = a ′ i for i = 1, . . . m.
If C is a class of structures, we denote by C [m -] the class of (R, a 1 , . . . a m ) where R ∈ C. Such type of labelling and embedding is a special case of the one mentioned above: we label by unary relations which take value 1 on exactly one element.

With these notions of labelling we may strengthen the notion of wqo. Let C be a class of relational structures. If m is a nonnegative integer, we say that C is m-wqo, resp., m --wqo, if C [m] , resp. C [m -] , is wqo. We say that C is very well-quasi-ordered, in short vwqo, if C is m-wqo for every integer m. Let P be a wqo, we say that C is P -wqo if C ⋅ P is wqo. With this notion, C is vwqo if and only if C is P -wqo for every finite poset P . We say that C is hereditarily wqo if C is P -wqo for every wqo P . We have to reserve this latter notion for classes of finite structures. Indeed, if R is the chain ω of the integers, the fact that {R} ⋅ P is wqo is equivalent to the fact that I(P ) is wqo. The example of Rado Rad is wqo but I(Rad) is not wqo. Thus, for infinite structures the appropriate notion is rather that of hereditarily bqo.

Examples 4.2. Trivially, every finite union of hereditarily wqo classes is hereditarily wqo; hence every finite subclass C of Ω µ is hereditarily wqo. More interestingly, if Chain is the class of finite chains, the class Chain ⋅A of finite chains labelled by a poset A can be identified with the set A * of finite words over the alphabet A equipped with the Higman ordering. The fact that Chain is hereditarily wqo is Theorem 3.18. More generally the collection of finite direct sums of finite chains is hereditarily wqo. In fact, the collection of finite series-parallel posets is hereditarily wqo (apply the full version of Higman's theorem, namely Theorem 3.17). More examples follow from the following two facts. Fact 1. If C is hereditarily wqo then its image by a free operator is hereditarily wqo.

In particular, the ages of structures freely interpretable by a chain are hereditarily wqo. These ages are the basic blocks in a construction of ages. Their properties are given in Subsection 5.1.

Fact 2. If C is a class of finite relational structures, C is hereditarily wqo if and only if ↓ C, its downward closure, is hereditarily wqo.

For an example, the class Tree of finite ordered structured trees is hereditarily wqo since by Kruskal's Theorem, the class Tree ⋅A of structured trees labelled by a wqo poset A is wqo. Hence, ↓ Tree is hereditarily wqo.

Given a hereditary class C, the relationship between the wqo character of C [m] and C [m -1 ] for various m is almost totally unknown. We have examples demonstrating that: (1) there are ages of graphs which are wqo but not 1 --wqo; (2) there are ages of graphs which are 1 --wqo but not 2 --wqo.

Examples 4.3. Regarding (2), the age of an infinite path is 1 --wqo but not 2 --wqo. The age of an infinite path consists of direct sums of finite paths. As seen in Example 3.20, it is wqo. The collection of finite paths with one label is isomorphic to the set of pairs (n, m) ∈ ω × ω such that n ≤ m, thus is wqo. Hence, from Lemma 3.21, the set of direct sums of finite paths with at most one label is wqo. With two labels, it is not wqo. Indeed, let P n be the n-vertex path (n ≥ 2) with two constants, one for each extremity of the path. Then the family of these labelled paths is an antichain. The example given at the end of Subsection 3.7 is an adaptation (experience shows that all antichains of graphs are built by an appropriate adaptation of this, but no theorem has emerged yet).

Regarding (1), if µ is a word on a 2-letter alphabet which is uniformly recurrent then the age of the graph G µ defined in Subsection 7 is wqo (this age is minimal prime by Theorem 7.16 and wqo by Theorem 7.23). If µ is uniformly recurrent and non periodic, then the age of G µ is not 1 --wqo. The fact that this age is not 1 --wqo follows essentially from the fact that the collection Fac(µ) of finite factors of µ is not wqo for the prefix ordering (α ≤ pref β if β = α.γ for some word γ). The argument for this fact is not difficult. If Fac(µ) was wqo, it will be a finite union of ideals. For each nonnegative integer n, the set Pref(µ, n) made of finite prefixes of µ ↾[n,∞[ is an infinite ideal of Fac(µ) for the prefix order. Any infinite ideal (for this ordering) of Fac(µ) is made of the prefixes of some infinite word, hence it does not contain an infinite proper ideal. Consequently, there are two distinct nonnegative integers n and m such that Pref(µ, n) = Pref(µ, m). From this follows that µ is eventually periodic, a contradiction.

The notion of very-well-quasi-ordered class appeared in 1972 in our note [START_REF] Pouzet | Un belordre d'abritement et ses rapports avec les bornes d'une multirelation[END_REF] (under the French name très belle classe). The following problems emerged at this time; up to now they are unsolved. Problems 9. Let C be a class of finite relational structures.

(1) Is it true that C ⋅ P is wqo for every wqo P whenever C [START_REF] Abdi | An example of Tateno disproving conjectures of Bonato-Tardif, Thomassé and Tyomkyn[END_REF] is wqo? whenever C [2 -] is wqo? (2) Let P be a wqo. Is it true that C ⋅ P is wqo whenever C ⋅ P ′ wqo for every finite P ′ ≤ P ? (3) Is it true that C is 1-wqo whenever it is m --wqo for every integer m? Note that the converse trivially holds.

For some categories of finite structures, the answer is negative. Kriz and Sgall [START_REF] Kříž | Well-quasiordering depends on the labels[END_REF] constructed for each integer n a concrete category of finite objects, say C, such that C labelled by an n-element chain is wqo and C labelled by a n + 1-element chain is not. More generally, they show by a clever construction that the set of countable ordinals γ such that there is some C such that C ⋅ γ ′ is wqo for every γ ′ < γ but is not wqo for γ ′ = γ is uncountable. But their objects are not relational structures.

4.2.

Well-quasi-ordering and finiteness of the kernel of a relational structure. The fact that the age of a relational structure R is wqo does not imply that R has a finite kernel -an example is an infinite path with a loop at the initial element of the path. With labelling, the situation is different. We have for an example Corollary 4.5 below (Corollaire p.6 in "Caractérisation combinatoire et topologique des âges les plus simples" [START_REF] Pouzet | Sur la théorie des relations[END_REF]). For the reader's convenience, we give a proof. It uses the characterization of the kernel recorded in Lemma 3.32.

Proposition 4.4. Let R ∶= (V, (ρ i ) i∈I ) be a relational structure made of finitely many relations. If the set Age(R) [1 -] ∶= {(S, a) ∶ S ∈ Age (R), a ∈ V (S)} is a finite union of ideals, then ker(R) is finite.

Proof. Suppose that ker(R) is infinite. We build a sequence (R ↾An , a n ) of elements of Age(R) [1 -] such that no two members of the sequence have a common extension belonging to Age(R) [1 -] . By construction, these members form an infinite antichain of Age(R) [1 -] , moreover if Age(R) [1 -] is an union of a family of ideals, no pair of distinct members of this sequence can be contained in one of these ideals. Hence this family of ideals must be infinite. To define the sequence, suppose (A n , a n ) be defined for n < m, pick a m ∈ V ∖ ⋃ n<m A m , select A given by Lemma 3.32 and set A m ∶= A ∪ ⋃ n<m A n .

According to Theorem 3.2 every wqo poset is a finite union of ideals, hence we get: Corollary 4.5. If the age of R is 1 --wqo then ker(R) is finite. Proposition 4.6. Let R ∶= (V, (ρ i ) i∈I ) be a relational structure made of finitely many binary relations and A ∶= Age(R). If every proper subage of A is hereditarily wqo then ker(R) is finite.

Proof. If ker(R) = ∅, there is nothing to prove. Otherwise, pick a ∈ ker(R). Define S ∶= R -a U 1 . . . U k where the U i 's are unary relations on V ∖ {a} such that the local isomorphisms of S extend by the identity on a to local isomorphisms of R. The structure S is hereditarily wqo and in particular 1 --wqo, hence by Corollary 4.5 its kernel ker(S) is finite. Since ker(R) = {a} ∪ ker(S), ker(R) is finite.

We do not know if the conclusion holds for relational structures which are not necessarily binary. That is: Problem 10. Is ker(R) finite whenever R is made of finitely many relations and every proper subage of Age R is hereditarily wqo?

A special instance of this problem is when R is made of finitely many relations and H(Age(R)) < ω 2 ; indeed in this case every proper subage of Age(R) is hereditarily wqo (Theorem 5.34). We will see in Subsection 5.4 that in this case the answer to our problem is positive. 4.3. Bounds of labelled wqo classes. The notions above allow to prove that some hereditary classes are wqo and have finitely many bounds. For example, we have: If instead of graphs, one considers directed graphs then to get the same conclusion with a similar proof would requires that C [START_REF] Albert | Simple permutations and pattern restricted permutations[END_REF] is wqo (and that C [2k] is wqo if C is made of binary structures each consisting of k binary relations). For directed graphs or multigraphs, one can do better: Lemma 4.8. Let C be a hereditary class of binary structures with a given finite arity. If the class C [1,1 -] of R ⋅ U ⋅ a, where R ∈ C, U is unary and a is a constant, is wqo then the cardinality of bounds of C is bounded.

Proof. We suppose that the cardinality of bounds of C is unbounded and we get a contradiction. Since C is wqo this is a finite union of ages (Corollary 3.11). Since the cardinality of bounds of C is unbounded, then the cardinality of bounds of some age A included into C is also unbounded. Choose A minimal with respect to set inclusion. It follows that every member of A embeds into some members of F orb(A) of arbitrarily large finite cardinality. Let B ∈ F orb(A). Let b ∈ V (B). Let E b (B) be the set of S ∈ A with the same base as B that coincide with B on V (B) ∖ {b}. We claim that E b (B) is nonempty. Indeed, B ↾V (B)∖{b} embeds into members of F orb(A) of arbitrarily large finite cardinality. For each S ∈ E b (B) and x ∈ V (S) set φ B (x) ∶= (B(b, x), B(x, b)) and φ S (x) ∶= (S(b, x), S(x, b)) and set

Good(S) ∶= {x ∈ V (S) ∶ φ B (x) = φ S (x)}.
Choose S such that the cardinality of Good(S) is maximum and set S B ∶= S ⋅ Good(S) ⋅ b. Since A [1,1 -] is wqo, we may extract a strictly increasing sequence

S Bn ∶= S n ⋅ Good(S n ) ⋅ b n of the set of S B for B ∈ Forb(A).
For each n, let f n be an embedding of S Bn into S B n+1 ; for m > n, let g n,m ∶= f m-1 ○ ⋅ ⋅ ⋅ ○ f n and x 0 ∈ V (S 0 ) ∖ Good(S 0 ) and x n+1 ∶= f n (x n ). We claim that there are n < m such that φ Bn (x 0 ) = φ Bm (x m ). Indeed, we may suppose that |B n | ≥ 3, hence the restriction of B n to {b n , x n } belongs to A which is wqo and thus there are only finitely many such restrictions. Since

|V (B n )| < |V (B m )|, we may pick u ∈ V (B m ) ∖ Range(g n,m ). Since B m ∈ Forb(A), B m ↾ (V (B m ) ∖ {u}) ∈ A, hence B m ↾ Range(g n,m ) ∈ A. Let S ′ n be its inverse image by g n,m . Then S ′ n ∈ E bn (B n ). Furthermore Good(S ′ n ) ⊇ Good(S n ) ∪ {x n }.
This contradicts the maximality of Good(S n ). If we deal with undirected loopless graphs, the hypothesis in Lemma 4.7 is weaker than in Lemma 4.8.

[m-1,m-1 -] of R ⋅ U 1 ⋅ ⋅ ⋅ U m-1 ⋅ a 1 ⋅ ⋅ ⋅ a m-1
, where R ∈ C, U 1 , . . . , U m-1 are unary and a 1 , . . . , a m-1 are m -1 constants, is wqo then the cardinality of bounds of C is bounded.

This result (with the stronger assumption that the labelling uses m constants instead of m -1) is in [START_REF] Pouzet | Un belordre d'abritement et ses rapports avec les bornes d'une multirelation[END_REF]; the technique used here in the proof of Lemma 4.8 allows us to reduce the number of constants by 1. As a consequence we have: Theorem 4.10. Provided that the signature µ is bounded, the cardinality of bounds of every hereditary and hereditarily wqo subclass of Ω µ is bounded.

With Theorem 4.10 and Lemma 3.12 we obtain: Corollary 4.11. A hereditary class C of finite structures with a finite signature is wqo whenever every age properly included in C is hereditarily wqo.

With the notion of hereditarily wqo classes, we obtain the following generalization of Lemma 3.21. Proposition 4.12. Let C be a class of binary structures of arity µ and R be a class of finite binary structures of arity µ. If C is wqo and R is hereditarily wqo then ∑R C is wqo. If in addition C is hereditarily wqo then ∑R C is hereditarily wqo.

Proof. The proof is an application of the minimal bad sequence technique of Nash-Williams. We consider finite ordered trees which are ramified in the sense that every non terminal vertex is the meet of two incomparable vertices. If T is a finite ordered tree and x ∈ T , we denote by T ↑x the restriction to ↑ x of the tree T ; if x < y, we set T (x, y) for the unique successor of x below y. We consider a class T of labelled ramified trees defined as follows. If x is a terminal vertex of T ∈ T, it is labelled by some member C x of C, if x is non terminal, the set T (x) of immediate successors of x in T is the domain of some R x ∈ R. If T and T ′ are two such trees, we set T ≤ T ′ if there is a 1-1 meet-preserving map f from T to T ′ such that the following holds: (a) for every vertex

x of T , f (x) is a terminal vertex of T ′ and C x is embeddable into C ′ f (x) if x is a terminal vertex of T ; (b) for every non terminal vertex x the map y ↦ T ′ (x, f (y)) from T (x) to T ′ (x) is an embedding of R x into R ′ f (x) . We set T ≺ T ′ if T ≤ T ′
↑x for some successor x of the root of T ′ . To each T ∈ T we associate F (T ) ∈ ∑R C by the following process defined inductively: if T has just one vertex, F (T ) is the label of that vertex, if T has more that one vertex, and r is the root of T , then F (T ) is the lexicographical sum ∑ x∈Rr F (T ↑x ) where for each x ∈ T (r), T ↑x is the restriction to ↑ x of the labelled tree T . One easily checks that the map F is order preserving and surjective, hence the image is wqo provided that T is wqo. If T is not wqo then it contains a bad sequence, and since ⪯ is well-founded it contains a bad sequence T 0 , . . . , T n , . . . which is minimal in the sense that if T ′ 0 , . . . , T ′ n , . . . is a sequence which coincides up to T n and T ′ n+1 ⪯ T n then this sequence is not bad. From this it follows that the set T ′ of T ′ such that T ′ ≺ T n for some n is wqo. Since C is wqo, only finitely many T n are reduced to their root. Let n such that T n has more than one vertex. Let r n be the root of T n and R rn ∈ R on the set of successors x of r n . Labelling each successor x by the labelled tree T n↑x yields the labelled relational structure R rn ∶= (R rn , (T n,↑x ) x∈Tn(rn) ). Since R is hereditarily wqo and T ′ is wqo, the collection of these labelled structures is wqo In particular there are n < m such that R rn ≤ R rm . But, as this can be easily checked, this implies T n ≤ T m , a contradiction. Suppose now that C is hereditarily wqo. Let P be a wqo. In order to prove that ∑R C⋅P is wqo, it suffices to replace C by Cz ⋅P in the above proof.

If R is a finite binary relational structure then the modules of R and of R ⋅ U 1 ⋅ . . . ⋅ U k , where U 1 , . . . , U k are unary relations, are the same. (Note that this does not hold with the notion of module of Fraïssé.) Hence, Theorem 3.23 admits the following refinement: A class C of finite binary structures with a finite signature which contains only finitely many prime members (up to isomorphy) is very well-quasi-ordered.

Moreover, from Proposition 4.12, we obtain the following result.

Theorem 4.13. A hereditary class C of finite binary structures with a finite signature which contains only finitely many prime members (up to isomorphy) is hereditarily well-quasi-ordered.

4.4.

Preserving wqo by adding a linear order. If the wqo character of some classes of structures can be preserved by adding unary predicates, there is no chance it will be preserved by adding binary predicates. An obvious reason is that contrarily to unary relations, the classes of binary relations are not necessarily wqo. Even if the binary relations that we add happen to form a wqo class, the resulting structures do not form a wqo in general. For example, the class of bichains is not wqo under embeddability [START_REF] Spielman | An infinite antichain of permutations[END_REF]. Still, I guess that it could be interesting to look at classes of structures whose wqo character is preserved by adding appropriately chosen linear orders. In that respect, We will mention the fact that the automorphism group of a countable homogeneous ordered structure is extremely amenable if and only if the structure has the Ramsey property [START_REF] Kechris | Fraïssé limits, Ramsey theory, and topological dynamics of automorphism groups[END_REF].

Problems 12.

(1) Let C be a wqo hereditary class of finite structures. Is it true that for every S ∈ C there is some linear order L S on V (S), the domain of S, such that the class of S ⋅ L S is wqo? (2) If an age A is wqo, is it true that Age(R ⋅ L) is wqo for some R such that Age(R) = A and a linear order L on V (R)?

A positive answer to the second problem yields a positive answer to the first. Indeed, if C is wqo then C is a finite union of wqo ages A i . Each A i is the age of some relational structure R i . If [START_REF] Albert | Simple permutations and pattern restricted permutations[END_REF] has a positive answer, there is some linear order L i such that Age(R i .L i ) is wqo. Then, C ′ ∶= ⋃ i=1,n Age(R i .L i ) is wqo. If for each S ∈ C one selects some L S such that S ⋅ L S ∈ C ′ , the resulting class being included into C ′ is wqo.

Note that if [START_REF] Abdi | An example of Tateno disproving conjectures of Bonato-Tardif, Thomassé and Tyomkyn[END_REF] has a positive answer, then we may associate to every member S ∈ A some linear order L S such that the class A ′ of S ⋅ L S is wqo. We do not know if the hereditary closure of A ′ contains a wqo hereditary subclass whose first projection is A. (This would hold were A a hereditarily wqo.) Also, observe that if the second problem has a positive answer, then for every R with Age(R) = A there is some linear order L on V (R) such that Age(R ⋅ L) is wqo. Indeed, let R 0 and L 0 such that Age(R 0 ) = A and A ′ ∶= Age(R 0 ⋅ L 0 ) is wqo. Let P be the free operator transforming each concatenat R ′ ⋅ L ′ to R ′ . By hypothesis, A is the image of A ′ hence, via colorred compactness of first order logic, for every R with Age(R) = A there is some L such that Age(R ⋅ L) ⊆ A ′ . Hence, Age(R ⋅ L) is wqo.

Ordered structures have peculiar properties. For instance, following similar steps as in the proof of Lemma 4.8, it can be shown that: Lemma 4.14. If the age of an ordered directed multigraph G is 2 --wqo then it has finitely many bounds. Consequently, there are only countably many hereditary classes of ordered graphs which are 2 --wqo.

Problem 13. How many hereditary classes of finite graphs are 2 --wqo?, 1 --wqo? How many hereditary classes of finite ordered graphs are 1 --wqo? 4.5. Uniform prehomogeneity. A relational structure R is uniformly prehomogeneous if for every finite set F of the domain V of R there is a finite superset F ′ of F whose cardinality is bounded by some function θ of the cardinality of F such that every local isomorphism of R defined on F extends to an automorphism provided that it extends to F ′ . This notion extends the notion of homogeneity. It was introduced by Pabion [START_REF] Pabion | Relations préhomogènes[END_REF] for relational structures with finite signature. For relational structures with possibly infinite signature, a closely related concept was defined in terms of maximal existential types and a necessary and sufficient condition for existence was given in [START_REF] Pouzet | Modèle universel d'une théorie n-complète: Modèle uniformément préhomogène[END_REF]. For our purposes here it suffices to say that the two notions coincide for structures with finite profile. Problem 14. Let R be a uniformly prehomogeneous structure. If Age(R) is wqo, is it true that (R, L) is uniformly prehomogeneous and Age(R, L) is wqo for some linear order L on V (R)?

It was proved independently in [START_REF] Pouzet | Modèle universel d'une théorie n-complète: Modèle uniformément préhomogène[END_REF] and [START_REF] Saracino | Model companions for ℵ0-categorical theories[END_REF] that: Theorem 4.15. A countable relational structure R such that Aut(M ) is oligomorphic is equimorphic to a countable R ′ which is uniformly prehomogeneous and furthermore Aut(R ′ ) is oligomorphic.

From the test of existence of uniformly prehomogeneous structures given in [START_REF] Pouzet | Modèle universel d'une théorie n-complète: Modèle uniformément préhomogène[END_REF] one has: Theorem 4.16. If an ideal A of finite relational structures is m --wqo for every nonnegative integer m then there is a uniformly prehomogeneous structure R with Age(R) = A.

Proof. Here is a sketch of the argument. Fix a nonnegative integer m. Let A(m) be the collection of (S, a 1 , . . . , a m ) where S ∈ A and a 1 , . . . , a m ∈ S. Quasi-ordered by embeddability, this set is wqo. Hence, it is a finite union of ideals. This is the test given in [START_REF] Pouzet | Modèle universel d'une théorie n-complète: Modèle uniformément préhomogène[END_REF] in terms of existential m-types for the existence of a uniformly prehomogeneous structure R with age A. Problem 15. If R is uniformly prehomogeneous and Age(R) wqo, is it true that Age(R) is m --wqo for every nonnegative integer m?

HEREDITARY CLASSES OF SMALL LENGTH AND AGES OF SMALL HEIGHT

This section is about hereditary classes of small length, that is, classes of height less than ω 2 . These classes extend classes of monomorphic, chainable, almost monomorphic and almost chainable structures. These former classes appeared first in the thesis of Fraïssé [START_REF] Fraïssé | Sur quelques classifications des systèmes de relations[END_REF]. They play a crucial role in the proof that the profile is non-decreasing (Theorem 6.1). We give an other illustration with the characterization by Cameron of monomorphic permutation groups. A careful examination of the notions of almost monomorphy and almost chainability allows extension of these properties to ages of almost multichainable structures. These ages are wqo, in fact hereditarily wqo, hence have finitely many bounds (Theorem 5.34 and Theorem 5.39), and have height less than ω ω (Theorem 5.42). The central result of this section is that ages of height less than ω 2 are ages of almost multichainable structures (Theorem 5.35). This allows us to characterize the members of the Cantor-Bendixson derivatives of finite rank of the space of subages of a given age as we did in Subsection 3.6 and also to prove the existence of jumps in the profile as presented in Section 6. 5.1. Height at most ω. Monomorphy and chainability. If P is a finite poset, o(P ) and h(P, I(P )) are equal to the cardinality of P . If P is an infinite poset and every initial segment is finite then P is countable and o(P ) = h(P, I(P )) = ω. Such a poset P is Jónsson. Jónsson posets were introduced by Oman and Kearnes [START_REF] Kearnes | Jónsson posets and unary Jónsson algebras[END_REF]. Countable Jónsson posets were studied and described in [START_REF] Pouzet | Sur la théorie des relations[END_REF][START_REF] Pouzet | From well-quasi-ordered sets to better-quasi-ordered sets[END_REF][START_REF] Assous | Jónsson posets[END_REF]. We recall Proposition 3.1 in [START_REF] Assous | Jónsson posets[END_REF].

Theorem 5.2. Let P be a countable poset. The following statements are equivalent.

(i) P is Jónsson; (ii) P is wqo and o(P ) = ω; (iii) P is level-finite, has height ω, and for each n < ω, there is m < ω such that each element of height at most n is below every element of height at least m.

Jónsson posets are in fact better-quasi-ordered (bqo) (see Subsection 3.4 for the definition). This fact, observed in [START_REF] Pouzet | Sur la théorie des relations[END_REF], has been significantly extended [START_REF] Carroy | From well to better, the space of ideals[END_REF].

We state and prove Lemma 4 of [START_REF] Oudrar | Minimal prime ages, words and permutation graphs[END_REF]:

Lemma 5.3. Every infinite well-founded poset P which is level-finite contains an initial segment which is Jónsson.

Proof. Observe that the intersection of every descending sequence (I α ) α∈κ of infinite initial segments I α of P is infinite. Indeed, let I ∶= ⋂ α∈κ I α . Fix n < ω and let P n be the n-level of P . We have [START_REF] Fraïssé | Theory of relations[END_REF] and [START_REF] Pouzet | Application de la notion de relation presque-enchaînable au dénombrement des restrictions finies d'une relation[END_REF].) The results obtained by Fraïssé and Frasnay [START_REF] Frasnay | Quelques problèmes combinatoires concernant les ordres totaux et les relations monomorphes[END_REF] in this area form one of the most important chapters of the theory of relations. We recall some basic facts that can be found in [START_REF] Pouzet | Application de la notion de relation presque-enchaînable au dénombrement des restrictions finies d'une relation[END_REF]. We focus on monomorphic classes since the properties of classes of higher complexity, such as those of almost chainable or almost multichainable structures, are extensions of monomorphic classes. Given a relational structure R and a nonnegative integer p, R is said to be p-monomorphic if its restrictions to finite sets of cardinality p are isomorphic. The structure R is monomorphic if it is p-monomorphic for every integer p.

I ∩ P n = ⋂ α∈κ (I α ∩ P n ).
There are eight types of monomorphic directed graphs, four made of reflexive directed graphs, four made of irreflexive graphs. For the reflexive ones, there are the chains, the reflexive cliques, the antichains, plus the 3-element oriented reflexive cycle. Whereas, for the irreflexive ones, there are the acyclic (oriented) graphs, the cliques, the independent sets, and the 3-element oriented irreflexive cycle. Other examples of monomorphic relational structures can be found in [START_REF] Krauss | Universally complete universal theories[END_REF].

Fraïssé gave a characterization of infinite monomorphic relational structures by means of his notion of chainability: a relational structure R ∶= (V, (ρ i ) i∈I ) is chainable if there is a linear ordering ≤ on V such that L ∶= (V, ≤) interprets R freely, that is, every local isomorphism of L ∶= (V, ≤) is a local isomorphism of R. Such a linear order ≤ is said to chain R. Chains are monomorphic, hence chainable relational structures are also monomorphic. The converse does not hold (as shown by a 3-element cycle). Fraïssé proved that it holds if the structure is infinite.

Theorem 5.4. An infinite relational structure is monomorphic if and only if it is chainable.

Fraïssé's proof, given for relational structures of finite signature, was based on Ramsey's theorem [START_REF] Ramsey | On a problem of formal logic[END_REF] and the compactness theorem of first order logic (in a weaker form, given by Fraïssé's "coherence lemma"). The extension to arbitrary signature requires an other application of the compactness theorem. Since Ramsey's theorem and the compactness theorem of first order logic are among the most important tools of the theory of relations, we give the proof idea (for a detailed proof, see Section IV, p. 295 of [START_REF] Pouzet | Application de la notion de relation presque-enchaînable au dénombrement des restrictions finies d'une relation[END_REF]). With Ramsey's theorem, Fraïssé proved the following:

Lemma 5.5. Every infinite relational structure with a finite signature has an infinite chainable restriction.

Proof. Let R be a relational structure and V be its domain. We suppose V infinite. Let A be a countably infinite subset of V . Let m be the maximum of the signature of R and let [A] m be the set of m-element subsets of A. Put a linear order ⪯ of type ω on A. Define an equivalence relation on

[A] m : say that two m-element subsets X, X ′ ∈ [A] m are equivalent if the unique order isomorphism of (X, ⪯ ↾X ) on (X ′ , ⪯ ↾X ′ ) is a local isomorphism of R.
Observe that the number of equivalence classes is finite. Apply Ramsey's theorem and get an infinite subset A ′ ⊆ A such that all elements of [A ′ ] m are equivalent. We may suppose that all m ′ -elements subsets of A ′ are equivalent for all m ′ ≤ m (if this is not the case, replace A ′ by A ′ minus its first m -1-elements). It follows that for V ′ ∶= A ′ , the restriction R ↾V ′ is chainable.

From Lemma 5.5, Fraïssé deduced that if R ∶= (V, (ρ i ) i∈I ) is a an infinite monomorphic relational structure then all its finite restrictions are chainable. Indeed, let I ′ be a finite subset of I. Then, the reduct

R ↾I ′ ∶= (V, (ρ i ) i∈I ′ ) is monomorphic. From Lemma 5.5, it contains an infinite induced structure (R ↾I ′ ) ′ which is chainable. Since R ↾I ′
is monomorphic, each finite substructure of R ↾I ′ is isomorphic to some finite substructure of (R ↾I ′ ) ′ hence is chainable. If X is a finite subset of V , the number of linear orders on X being finite, there is a linear order ≤ X which chains all the finite reducts of R ↾X , thus R ↾X is chainable as claimed. The application of the compactness theorem of first order logic is as follows. For each finite subset X of V , let U(X) be the set of binary relation ρ on V such that ρ ↾X is a linear order chaining R ↾X . Each U(X) is a nonempty closed subset of the space ℘(V × V ) made of binary relations on V . Since U(X) ∩ U(X ′ ) ⊇ U(X ∪ X ′ ) for every finite X, X ′ ⊆ V , the intersection of finitely many of these closed sets is nonempty. Since ℘(V × V ) is compact, the intersection of all these closed sets is nonempty. Every member of this intersection is a linear order which chains R. With that fact, the proof of Theorem 5.4 is complete.

◻

In his thesis, Frasnay 1965 ( [START_REF] Frasnay | Quelques problèmes combinatoires concernant les ordres totaux et les relations monomorphes[END_REF]) obtained the following result:

Theorem 5.6. For every integer m there exists an integer d such that every d-monomorphic relational structure R whose maximum of the signature is at most m and domain infinite or sufficiently large is chainable.

The least integer d in the statement of Theorem 5.6 above is the monomorphy threshold, d m . An upper bound of d m is in Frasnay's thesis. In 1990, Frasnay [START_REF] Frasnay | Détermination du degré optimal dm de monomorphie pour les structures relationnelles au plus m-aires[END_REF], using a result of Hodges et al [START_REF] Hodges | Possible orderings of an indiscernible sequence[END_REF] and his theory of permuted chains, obtained the exact value:

d 1 = 1, d 2 = 3, d m = 2m -2 for m ≥ 3.
For a detailed exposition see [START_REF] Fraïssé | Theory of relations[END_REF] Chapter 13, notably p. 378.

Frasnay deduced the following consequence (cf. 13.1.2, Corollaire, p. 519 in [START_REF] Frasnay | Quelques problèmes combinatoires concernant les ordres totaux et les relations monomorphes[END_REF]).

Theorem 5.7. The age of a chainable structure with a finite signature has finitely many bounds.

If we are only concerned with the existence of d m and not its numerical values, we can derive it from Theorem 5.7. Indeed, with Ramsey's theorem, for every integer p, all the p-element restrictions of a pmonomorphic member S of Ω µ are chainable provided that the cardinality of S is large enough. Hence, if p is the maximum of the cardinality of the bounds of the class Chain µ of finite chainable structures of signature µ, S is chainable. Thus, it suffices to prove that the cardinality of bounds of Chain µ is bounded. If the signature µ is finite and m is the maximum of the arity, Chain µ is the union of finitely many ages of chainable structures. Since a finite union of hereditary classes with finitely many bounds has finitely many bounds, Chain µ has finitely many bounds. This fact was obtained by Jean [START_REF] Jean | Relations monomorphes et classes universelles[END_REF], who also proved that the class Monom µ of monomorphic structures of signature µ has only finitely many bounds (these two classes are universal in the sense of Tarski).

Frasnay obtained Theorem 5.6 throught a deep analysis of the structure of permutation groups attached to chainable structures. Theorem 5.7, which is an equivalent form, follows from the fact that Chain µ is hereditarily wqo. Indeed, from that, the age of each chainable structure is hereditarily wqo, hence, by Theorem 4.10, the cardinality of its bounds is bounded. The price to pay with this simplification is the absence of numerical estimates.

(R ↾V ′ ) = Age(R) = C for every infinite subset V ′ of V . If I ′ is any subset of I, this property holds for the reduct R ↾I ′ . If I ′ is finite, Lemma 5.5 ensures that R ↾I ′ has an infinite chainable restriction R ↾I ′ ↾V ′ . Hence, Age(R ↾I ′ ↾V ′ ) = Age(R ↾I ′ ). It follows that R ↾I ′ is chainable.
The compactness argument given after the proof of Lemma 5.5 yields that R is chainable.

An illustration: Cameron's theorem on monomorphic permutation groups.

A group G of permutations on a set V is monomorphic (or set-homogeneous) if for every integer n and two n-element subsets A and B of V there exists an element σ of G that maps A to B (said differently, the group G acts transitively on subsets of V of a fixed finite cardinality). Cameron [START_REF] Cameron | Transitivity of permutation groups on unordered sets[END_REF] proved that on a countable set there are essentially five monomorphic closed groups (for the definition of closed group, see Subsection 1.4.6). In order to present his result, let us recall three well known structures associated to a chain C ∶= (V, ≤).

• The betweeness relation

B C ∶= (V, b C ) associated to C, where b C is the set of triples (x 1 , x 2 , x 3 ) such that either x 1 < x 2 < x 3 or x 3 < x 2 < x 1 . • The circular order T C ∶= (V, t C ) associated to C, where t C is the set of triples (x 1 , x 2 , x 3 ) such that x σ(1) < x σ(2) < x σ(3) for some circular permutation σ of {1, 2, 3}. • The betweeness relation D C ∶= (V, d C ) associated to the circular order, where d C is the set of quadruples (x 1 , x 2 , x 3 , x 4 ) such that x σ(1) < x σ(2) < x σ(3) < x σ(4) or x σ(4) < x σ(3) < x σ(2) < x σ(1)
for some circular permutation σ of {1, 2, 3, 4}. By construction, these three structures are chainable by C. Furthermore, if C is isomorphic to the chain of rational numbers, these three structures are actually homogeneous.

Here is Cameron's theorem.

Theorem 5.9. [START_REF] Cameron | Homogeneous permutations. Permutation patterns (Otago[END_REF] A monomorphic closed group on a countable set is isomorphic, as a permutation group, to one of the following automorphism groups: (a) S(Q), the full symmetric group on the set of rationals, viewed as the automorphism group of the equality relation; (b) Aut(Q), the automorphism group of the chain of rational numbers; (c) Aut(B Q ), the automorphism group of the betweeness relation associated to the chain of rational numbers;

(d) Aut(T Q ), the automorphism group of the circular order associated with the chain of rational numbers; (e) Aut(D Q ), the automorphism group of the betweeness relation associated to the circular order on the rationals.

In addition to the original paper of Cameron [START_REF] Cameron | Transitivity of permutation groups on unordered sets[END_REF], there are two research papers [START_REF] Higman | Homogeneous relations[END_REF], [START_REF] Pouzet | Application de la notion de relation presque-enchaînable au dénombrement des restrictions finies d'une relation[END_REF] and Chapter 13 in Fraïssé's book [START_REF] Fraïssé | Theory of relations[END_REF] giving a proof, as well as a presentation in [START_REF] Laflamme | Siblings of an ℵ0-categorical relational structure[END_REF].

We give some hints for a proof. We rely on [START_REF] Laflamme | Siblings of an ℵ0-categorical relational structure[END_REF].

Clearly, the groups mentioned in the theorem above are monomorphic and closed. The proof of the converse has three parts. Let G be a closed monomorphic group. Since G is closed, G is the automorphism group of a relational structuture R. If G is monomorphic, R is monomorphic. From Theorem 5.4, R is chainable. This is the first part. Now among the chains chaining R we claim that one is isomorphic to the chain of rational numbers. Indeed, let C be a chain chaining R. According to (1) of Lemma 2.18, there is a free operator P transforming C to R. Let R ′ be the image by P of the chain of rational numbers. Then R ′ and R have the same age. One can show that R ′ is uniformly prehomogeneous. This R ′ is unique up to isomorphism fot its age. Since R is homogeneous, it is uniformly prehomogeneous, hence R and R ′ are isomorphic (see 2.5 Lemme de préservation, p.320 [START_REF] Pouzet | Application de la notion de relation presque-enchaînable au dénombrement des restrictions finies d'une relation[END_REF]). This is the second part.

The third part is more subtle. It relies on properties discovered by Frasnay [START_REF] Frasnay | Quelques problèmes combinatoires concernant les ordres totaux et les relations monomorphes[END_REF][START_REF] Frasnay | Chainable relations, rangements and pseudorangements[END_REF] of pairs of chains chaining the same structure. Underlying this are the notions of group sequences, indicative sequences and results of Frasnay that we present briefly (according to the presentation in [START_REF] Laflamme | Siblings of an ℵ0-categorical relational structure[END_REF], Subsection 3.2, p. 100).

Let R be a chainable relational structure with domain V , and C be a chain (with the same domain V ) chaining R. Let n be an integer such that n ≤ |V | and let A ∶= a 1 < C ⋅ ⋅ ⋅ < C a n be a n-element subset of V ordered by C. The set of permutations σ of n ∶= {1, . . . , n} such that the map a i → a σ(i) is an automorphism of R ↾A forms a group. Since C chains R, this group is independent of the n-element set A and we denote it by Aut n (R, C). The sequence of these groups is called the group-sequence of the pair (R, C).

To a bichain B ∶= (V, C, C ′ ) we associate a sequence of permutations groups, called the indicative sequence of the bichain. Let n be a positive integer not larger than the cardinality of V , and let A be a n-element subset of V . Order A by C into the sequence a 1 < C ⋅ ⋅ ⋅ < C a n . Let σ be the unique permutation of n which reorders it into a σ(1) < C ′ ⋅ ⋅ ⋅ < C ′ a σ(n) in the order of C ′ . The collection of these permutations σ for n fixed, A belonging to all the n-element subsets of V , generates a subgroup Ind n (B) of S(n), called the n th indicative group of B. The sequence of these indicative groups is the indicative sequence of B. We can now recall the following result of Frasnay ([82], p. 263).

Lemma 5.10. The indicative sequence of a bichain B ∶= (V, C, C ′ ) whose components have no extreme elements is the group-sequence of one of the five relations defined in Theorem 5.9.

Since for every embedding φ of R into R, the inverse image of the chain C by φ provides a chain C ′ chaining R, automorphisms of R are automorphisms of one of the five relations above. The proof of the theorem follows.

Almost chainability, almost monomorphy. The notions of almost chainable and almost monomorphic

structures extend the notions of chainability and monomorphy. Results of Subsection 5.1, notably Theorem 5.7, extend to these structures (see [START_REF] Fraïssé | Interprétabilité d'une relation par une chaîne[END_REF][START_REF] Fraïssé | Sur une classe de relations n'ayant qu'un nombre fini de bornes[END_REF][START_REF] Pouzet | Relation minimale pour son âge[END_REF][START_REF] Pouzet | Application de la notion de relation presque-enchaînable au dénombrement des restrictions finies d'une relation[END_REF]). These results are presented below, but first we mention the profile, which emerged on the occasion of research on almost monomorphy and is examined in Section 6. Here, we just recall the definition. The profile of a class C of finite relational structures is the function φ C that counts for every integer n the number φ C (n) of members of C on n elements, isomorphic members being identified. The profile of a relational structure R, denoted by φ R , is the profile of its age. Definitions 5.11. Let R ∶= (V, (ρ i ) i∈I ) be a relational structure and K be a subset of V . The relational structure R is K-monomorphic if for every nonnegative integer n and every A, A ′ ∈ [V ∖ K] n there is an isomorphism from R ↾A onto R ↾A ′ which extends by the identity on K to an isomorphism of

R ↾A∪K onto R ↾A ′ ∪K . If L ∶= (V ∖K, ≤) is a chain, R is (K, L)-chainable if every local isomorphism of L, once extended by the identity on K, is a local isomorphism of R. The structure R is K-chainable if it is (K, L)-chainable for some chain L. It is almost monomorphic, resp. almost chainable, if it is K-monomorphic, resp. K- chainable for some finite subset K of V .
It is immediate to see that a relational structure R ∶= (V, (ρ i ) i∈I ) is K-chainable if and only if R is freely interpretable by a structure S ∶= (V, ≤, (a k ) k∈K ) where ≤ is a linear order on V , K is an initial segment of the chain (V, ≤) and the a k 's are constants defining K. This fact allows to apply (2) of Lemma 2.18. It yields: Lemma 5.12. If R is almost chainable then every R ′ with Age(R ′ ) included into Age(R) is almost chainable.

We will see in Theorem 5.16 that the notions of K-monomorphy and K-chainability coincide when V ∖K is infinite. This requires Ramsey's theorem and the compactness theorem of first order logic. But, before, it is worth recording the following result. It has some importance and requires no tools. Theorem 5.13. Let R be a relational structure and K be a subset of

V (R). If V (R) ∖ K is infinite and R is K-monomorphic (or K-chainable), then φ R is non-decreasing, that is, φ R (n) ≤ φ R (n + 1) for every integer n. Furthermore, max{φ R (n) ∶ n ∈ N} ≤ 2 |K| , if K is finite.
Proof. We relax the condition V (R) ∖ K infinite, supposing only that n < |V (R) ∖ K| and prove that φ R (n) ≤ φ R (n + 1). Let (τ i ) i≤φ R (n) be the types of the n-element restrictions of R. For each τ i , let F i be such that R ↾F i has type τ i and the cardinality of

K ∩ F i is minimum. Let x i ∈ V (R) ∖ (K ∪ F i ).
We claim that the restrictions R ↾F i ∪{x i } are pairwise non-isomorphic, proving that φ R (n) ≤ φ R (n + 1). If the claim does not hold, there are i / = j such that the restrictions R ↾F i ∪{x i } and R ↾F j ∪{x j } are isomorphic. Without loss of generality, we may suppose that

|K ∩ F i | ≥ |K ∩ F j |. Let f be an isomorphism from R ↾F i ∪{x i } onto R ↾F j ∪{x j } . Let y i ∶= f (x i ). Then R ↾F i and R ↾F j ∪{x j }∖{y i } are isomorphic. If y i / ∈ K then since R is K-monomorphic, R ↾F j ∪{x j }∖{y i } and R ↾F j are isomorphic, hence R ↾F i and R ↾F j are isomorphic. This is impossible. Hence y i ∈ K. But then f (F i ) ⊆ F j ∪ {x j } ∖ {y j }, hence |f (F i ) ∩ K| < |F j ∩ K| ≤ |F i ∩ K|,
contradicting our choices of F i and F j . Finally, note that if two n-element subsets coincide on K then the restrictions to these subsets are isomorphic, hence φ R (n) ≤ 2 |K| . From Ramsey's theorem, Fraïssé deduced the "lemma of the almost chainable restriction". Lemma 5.14. Let R be a relational structure with domain V and K be a subset of V . If V is infinite, K is finite and the signature of R is finite then there is a infinite subset

V ′ of V containing K on which the restriction R ′ ∶= R ↾V ′ is K-chainable.
The proof is a straightforward adaptation of the proof of Lemma 5.5. There is a "dual version" of this result (see [START_REF] Pouzet | Relation minimale pour son âge[END_REF], III.2.2.3, p.326). It requires less than the finiteness of the signature. Lemma 5.15. Let R be a relational structure with domain V and V ′ be a set containing V . Let C ∶= (V ′ ∖ V, ≤) be a chain. If Age(R) is level-finite, then there is a relational structure R ′ extending R on V ′ which is (V, C)-chainable and which has the same age as R.

The proof is based on Lemma 5.14 and the compactness theorem of first order logic. The use of this latter result is easier through the Diagram Method of Robinson: give names to the elements of V , use axioms to describe the diagram of R and C and express the (V, C)-chainability of finite reducts. Use Lemma 5.14 to get the consistency of those axioms. Obtain an extension of R and restrict it to V ′ . Use the fact that Age(R) is level-finite to prove that this restriction has the same age as R.

The analog of Theorem 5.4 also follows from Lemma 5.14 and compactness.

Theorem 5.16. Let R be a relational structure with domain V and K be a subset of

V . If V ∖ K is infinite, then R is K-monomorphic if and only if it is K-chainable.
Proposition 5.17. If an infinite relational structure R is almost chainable then there is a least subset (for set inclusion) K for which it is K-chainable. This set K is the kernel ker(R) of R.

Proof. Let R be K-chainable with K finite. Since V (R) ∖ K is infinite then for every x ∈ V ∖ K, Age(R -x ) = Age(R), hence ker(R) ⊆ K. Conversely, if A is any finite subset of V ∖ ker(R) then, since K is finite, there is some local isomorphism of R mapping A on some subset A ′ disjoint from K ∪ A and fixing ker(R) pointwise (Lemma 3.33). Since

R ↾K∪A ′ is K-chainable, R ↾ker(R)∪A ′ is ker(R)-chainable, hence R ↾ker(R)∪A is ker(R)-chainable. Via the Compactness Theorem of First Order Logic, R is ker(R)- chainable.
The existence of a least K was discovered by Fraïssé; eventually, it leads to the notion of kernel of an arbitrary relational structure. Proposition 5.18.

(1) If the signature of a countable relational structure R is finite then V (R) is the union of an increasing sequence (V n ) n∈N of countable subsets such that each restriction R ↾Vn is almost chainable.

(2) Every level-finite and infinite age A is the union of an increasing chain of ages of infinite almost chainable structures.

Proof. Let R be countable with A ∶= Age(R).

(1) Let (x n ) n∈N be an enumeration of V (R). Define a sequence

(K n , V n ) n∈N such that K n ∶= {x 0 , . . . , x n-1 } ⊂ V n ⊆ V (R), V n is infinite, R ↾Vn is K n -chainable and V n+1 ∖ K n+1 ⊆ V n . If (K n , V n )
is defined, apply Lemma 5.5 to R ↾Kn∪{x n+1 } and define (K n+1 , V n+1 ).

(2) Apply Lemma 5.15. Get some extension R ′ with Age(R ′ ) = A which is V (R)-chainable. Enumerate K by a sequence x 0 , . . . , x n , . . .

; set K 0 ∶= ∅, K n ∶= {x 0 , . . . , x n-1 } for n ≥ 1. Then R n ∶= R ↾Kn∪(V (R ′ )∖K) is K n -chainable.
The sequence of the ages of R n is increasing and their union covers Age(R).

An infinite hereditary class of finite structures is well-founded with respect to embeddability, but not necessarily level-finite. If it is level-finite, then Lemma 5.3 applies. This yields: Proof. (a) Since the signature µ is finite, C is level-finite, hence by Lemma 5.3 it contains a Jónsson hereditary subclass. This class has ordinal length ω, hence by Theorem 5.8, this subclass is the age of a chainable relational structure.

(b) By definition, for each chainable structure R there is some chain C chaining R. Since R is freely interpretable by C there is a free operator P such that P(C) = R. The number of free operators transforming binary relations into relational structures of signature µ (µ finite) is finite (cf. [START_REF] Fraïssé | Cours de Logique Mathématique, tome 1, Relation et formule logique[END_REF] Ch.4, 4.3.4, p. 90) hence the number of ages of chainable structures of signature µ is finite.

(c) Since µ is finite, Ω µ is level-finite hence, by Proposition 2.13, Id(Ω µ ) is topologically closed in ℘(Ω µ ).
The set of finite ages included in Ω µ is open in Id(Ω µ ). Indeed, since µ is finite, if C is a finite age it has finitely many bounds, and thus C is an isolated point of Id(Ω µ ) (the set of ages containing no bound of C and containing T such that ↓ T = C is equal to {C}). From this, the set of infinite members of Id(Ω µ ) is topologically closed, hence it is closed under intersection of totally ordered families. Apply Zorn's lemma to obtain that every infinite member is above some minimal one.

(d) Let C be an infinite minimal age containing T . Let R be such that Age(R) = C. Let K be a subset of the base V of R such that R ↾K is isomorphic to T . According Lemma 5.5, there is an infinite subset

V ′ of V containing K on which the restriction R ′ ∶= R ↾V ′ is K-chainable. Then Age(R ′ ) = C.
Remark 5.20. The conclusion of (a) holds also if C is level-finite. But, without some hypothesis, the conclusion of (a) may be false. There are hereditary classes containing arbitrarily large chainable finite structures with no age of an infinite chainable structure included into it. For an example, consider the class of structures of the form R ∶= (V, (ρ n ) n∈N ) where V is finite and for some n(V ) ≥ |V |, ρ n(V ) is a linear order on V , all ρ n are equal to ρ n(V ) for n ≤ n(V ) and equal to the empty binary relation for n > n(V ).

Theorem 5.21. The age of an almost chainable structure is hereditarily wqo. If the signature is bounded, then the cardinality of its bounds is bounded.

Proof. Let R ∶= (V, (ρ i ) i∈I ) be an almost chainable relational structure. Let K be a finite subset of V and ≤ be a linear order on V ∖ K witnessing that R is K-chainable. Let S ∶= (V, ρ, a 1 , . . . a k ) be the relational structure where ρ is the linear order on V extending the linear order ≤ on V ∖ K and putting all elements of K before those of V ∖ K, and a 1 , . . . , a k are constants defining K. We claim that Age(S) is hereditarily wqo. Indeed, any map f from a finite subset F of V into a poset P yields a pair of words, the first with domain included in K the second included in V ∖ K. If P is wqo, then from Higman's theorem, the set of these maps is wqo proving our claim. Now, since R is freely interpretable by S, Age(R) is hereditarily wqo. The second part of the Theorem is a consequence of Theorem 4.10.

The finiteness of the number of bounds was obtained in [START_REF] Fraïssé | Sur une classe de relations n'ayant qu'un nombre fini de bornes[END_REF] by reducing it to the finiteness of bounds of chainable structures. The proof occupies the whole paper.

We conclude this subsection by looking at several parameters of a hereditary classe C of finite structures.

(1) The number of infinite ages included in C. If C is level-finite this is the cardinality of the set Id (1) (C) of non isolated points of the topological space Id(C).

(2) The least number of chains needed to cover C, called the covering number of C and denoted by cov(C).

(3) The supremum sup φ C of the profile of C, that is the supremum of the cardinalities κ such that there are κ pairwise non-isomorphic members of C on the same number of elements. When κ is finite, C is said to be finimorphic, a notion invented by Fraïssé. (It was present in his 1968 graduate course of the theory of relations; see also [START_REF] Fraïssé | Interprétabilité d'une relation par une chaîne[END_REF].) (4) The ordinal length o(C) when C is wqo. Like every ordinal, the ordinal o(C) can be written o(C) = ω ⋅ β + p for a unique pair (β, p) of ordinals such that p < ω. The ordinal β, denoted by 1 ω ⋅ o(C), is the quotient of C by ω.

Theorem 5.22. Let A be an infinite ideal made of finite structures. Then the following properties are equivalent:

(i) A is the age of an almost chainable structure;

(ii) H(A) = ω + p where p is the largest integer such that there is an increasing sequence A 0 ⊂ A 1 ⊂ ⋅ ⋅ ⋅ ⊂ A p = A of infinite ages; (iii) the number of infinite ages included in A is finite. When one of these conditions holds, the number of infinite ideals included in A is |Id (1) (A)| and satisfies:

(5.1) |Id (1) (A)| = 1 ω ⋅ o(A) = sup φ C = cov(A).
Proof. (i) ⇒ (iii). Let R ∶= (V, (ρ i ) i∈I ) be an almost chainable structure with age A. Then R is freely interpretable by S ∶= (V, ≤, a 1 , . . . , a ℓ ) where ≤ is a linear order on V and a 1 , . . . , a ℓ are constants defining an initial segment of the chain (V, ≤). Let P be a free operator transforming S to R. According to (2) of Lemma 2.18 each age included in Age(R) is the image by P of an age included in Age(S). Trivially, the number of infinite ages included in the age of S is at most 2 ℓ . It follows that Age(R) contains only finitely many ages.

(iii) ⇒ (ii). This is immediate.

(ii) ⇒ (i). According to Theorem 2.8, A contains a countably infinite ideal, say A 0 . The height of A 0 , if it exists, is at least ω. Since H(A) = ω + p, the set of countable ages containing A 0 and included in A has height at most p + 1. Hence, this set contains a maximal member for set inclusion, say A ′ . I claim that A ′ = A. Otherwise, since A ∖ A ′ is up-directed and A ′ is countable we may, as in the proof of Theorem 2.8, build a countable ideal properly containing A ′ , contradicting its maximality. Since A ′ is countable, this is the age of some relational structure R ∶= (V, (ρ i ) i∈I ). The chains of infinite subages of A have length at most p + 1, hence according to (3) of Lemma 3.31, ker(R) is finite. Let I ′ be a finite subset of I. According to (1) of Proposition 5.18, V is the union of an increasing sequence (V n ) n∈N of countable subsets such that each restriction R I ′ ↾Vn is almost chainable. Since for each chain (X n ) n∈N of infinite subsets of V the sequence (Age(R ↾Xn )) n∈N is stationary, the sequence (Age(R I ′ ↾Xn )) n∈N is stationary too. Hence, Age(R

I ′ ) is almost chainable. Since ker(R I ′
) ⊆ ker(R), there is some finite

I ′ 0 such that ker(R I ′ ) = ker(R I ′ 0 ) for all finite I ′ containing I 0 . Hence, each R I ′ is K-chainable for K ∶= ker(R I ′ 0 ).
Via the compactness theorem of first order logic, R is K-chainable, hence almost chainable.

We prove that if (iii) holds, that is, if |Id (1) (A)| is finite, then the following inequalities hold:

(5.2) |Id (1) (A)| ≤ 1 ω ⋅ o(A) ≤ sup φ A ≤ cov(A) ≤ |Id (1) (A)|.
(a) cov(A) ≤ |Id (1) (A)|. Indeed, let A 0 , . . . , A d-1 be an enumeration of Id (1) (A) in a non-decreasing order. For i < d, let

X i ∶= A i ∖ ⋃ j<i A j . Then X i is nonempty (A i is join irreducible) and totally ordered (A i is almost chainable). Since the X i 's cover A into d chains, cov(A) ≤ d. (b) sup φ A ≤ cov(A). This is trivial. (c) 1 ω ⋅ o(A) ≤ sup φ A . Since A is infinite, it has no largest element, hence o(A) is a limit ordinal, that is, o(A) = ω.
α for some ordinal α. Since the number of infinite ages included in A is finite, α is finite. Let L be a linear extension of A with order type ω.α. Write it L = L 0 + ⋅ ⋅ ⋅ + L i + ⋅ ⋅ ⋅ + L α-1 with each L i having order type ω. We claim that we may select τ i ∈ L i such that all τ i 's have the same cardinality, say n. This proves that φ

A (n) ≥ α = 1 ω ⋅ o(A).
The claim holds because for each i < α there is an integer n i such that for each n ≥ n i some τ i with cardinality n belongs to L i (to see that note that each L i is a convex subset of A and this subset is wqo).

(d) |Id (1)

(A)| ≤ 1 ω ⋅ o(A). Proceed by induction on α ∶= 1 ω ⋅ o(A). If α = 1, then o(A) = ω, A is Jónsson, |Id (1)
(A)| = 1, and the inequality holds. If α > 1 then as above let L be a linear extension of A with order type ω.α. Write it L = L 0 + ⋅ ⋅ ⋅ + L i + ⋅ ⋅ ⋅ + L α-1 with each L i having order type ω. Apply induction on

L 0 + ⋅ ⋅ ⋅ + L i + ⋅ ⋅ ⋅ + L α-2 and L α-1 .
For a hereditary class, an extra hypothesis is needed and the result is a bit less precise. Theorem 5.23. Let C be a hereditary class of finite structures with a finite signature. Then the following properties are equivalent:

(i) each age A ⊆ C is the age of an almost chainable structure;

(ii) C is a finite union of ages of almost chainable structures;

(iii) C is coverable by finitely many chains;

(iv) C is finimorphic; (v) H(A) < ω ⋅ 2 for every A ∈ Id(C); (vi) o(C) < ω 2 ;
(vii) C contains only finitely many infinite ages.

Proof. We prove (i) ⇒ (ii) ⇒ (iii) ⇒ (iv) ⇒ (v) ⇒ (i) and (iii) ⇒ (vi) ⇒ (vii) ⇒ (i).

(i) ⇒ (ii). First, C is wqo. Otherwise, by Lemma 3.12, C contains an age with infinitely many bounds in C. This is impossible since each age included in C is the age of an almost chainable structure and such ages have only finitely many bounds. Since C is wqo, it is a finite union of ages and these ages are ages of almost chainables structures, as required.

(ii) ⇒ (iii). Each age of an almost chainable relation is coverable by finitely many chains.

(iii) ⇒ (iv). This is trivial.

(iv) ⇒ (v). If C contains a sequence A 0 ⊂ A 1 ⋅ ⋅ ⋅ ⊂ A n of infinite ages, then max φ An ≥ n + 1.
(v) ⇒ (i) and (vii) ⇒ (i). Let A be an infinite age. According to Proposition 5.18, A is the union of an infinite increasing sequence of ages of infinite almost chainable structures. If, according to (vii), A contains only finitely many infinite subages or, according to (v), H(A) < ω ⋅ 2, the sequence is stationary. Thus, A is the age of an almost chainable structure.

(iii) ⇒ (vi). If a poset is a finite union of well ordered chains, the ordinal length is majorized by the Hessenberg sum of these ordinals (Carruth [START_REF] De Jongh | Well-partial orderings and hierarchies[END_REF]). Each chain in C is either finite or has order type ω, so the ordinal length is at most ω ⋅ +n.

(vi) ⇒ (vii). This is essentially a property of posets. Suppose that C is a poset with o(C) = ω ⋅ m + n for m, n < ω, and prove that C contains at most m infinite ideals with no largest element. The proof is by

induction on o(C). If o(C) < ω, there is nothing to prove. If o(C) = ω + n, then C minus a finite set is Jónsson, hence it contains only one infinite age. If o(C) ≥ ω ⋅ 2, there is an initial segment C ′ such that o(C ′ ) = ω.(m -1) and o(C ∖ C ′ ) = ω + n. Apply induction to C ′ .
Note that without the finiteness of the signature, condition (i) is too weak to imply (vii). Indeed, let C ∶= Age(R) ∪ Bound(Age(R)) where R ∶= (N, (ρ i ) ∈N ), each ρ i being the complete graph on the set N. With respect to embeddability, the set C is the union of a chain, namely Age(R), and an antichain, namely Bound(Age(R)). This latter set is infinite, it contains each B m ∶= ({0, 1}, (β n ) n∈N ) where β n is the empty graph for n / = m and β m is the complete graph on {0, 1}. We introduce first the product of a relational structure by a chain, from which multichainable structures are a special case. 5.4.1. Product of a relational structure by a chain and multichainability. Let S ∶= (W, (τ i ) i∈I ) be a relational structure of signature µ ∶= (m i ) i∈I and L ∶= (D, ≤) be a chain. A relational structure R of signature µ is a product of S and L if: (a) the domain of R is D × W ; (b) for every x ∈ D the map y → (x, y) is an isomorphism from S to R ↾{x}×W ; (c) for each local isomorphism f of L, the map (f, 1 W ), which maps (x, y) to (f (x), y), is a local isomorphism of R. If W is finite, the relational structure R is said to be multichainable, the cardinality of W is a width of R.

This notion was introduced in [START_REF] Pouzet | Sur la théorie des relations[END_REF] and published in [START_REF] Pouzet | Relation minimale pour son âge[END_REF]. In that paper, it appears under the name of D-multiple of R (see IV.3.3.2 ). It appears also in [START_REF] Pouzet | The profile of relations[END_REF], p.264. ( In this reference R is defined on the direct product V × D. We denote by S ⊗ L such a product (despite the fact that S and L do not determine it). If |W | = 1, L chains this product. Hence R is chainable.

We may note that the age of a product S ⊗ L is inexhaustible if L is infinite. Several properties involving the notion of inexhaustibility are stated in [START_REF] Pouzet | Relation minimale pour son âge[END_REF] (for example, Lemme fondamental IV.3.3.5, Corollaire IV.3.3.6. Proposition IV.3.3.7). We extract the following result from Lemme fondamental IV.3.3.5 and Proposition 4.21, p. 266, of [START_REF] Pouzet | The profile of relations[END_REF].

Proposition 5.24. Let B be an inexhaustible age and S be a relational structure such that Age(S) ⊆ B. If B is level-finite then for every infinite chain L there is some product S ⊗ L such that Age(S ⊗ L) ⊆ B.

Proof. This result extends (a) of Proposition 5.19. It uses the same tools, namely Ramsey's theorem and the compactness theorem of first order logic. We suppose first that S ∈ B. Since B is inexhaustible there is a structure R on the direct product N × V (S) such that for n ∈ N the map y → (n, y) is an isomorphism from S to R ↾{n}×V (S) and Age(R) ⊆ B. Let µ ∶= (m i ) i∈I be the signature of R. From Ramsey's theorem, we obtain that for every finite subset I ′ ⊆ I there is an infinite subset N I ′ of N such that the reduct R ↾I ′ restricted to N I ′ × V (S) is a product of the reduct S ↾I ′ and the chain (N I ′ , ≤). Being level-finite, B is reduct-closed by (a) of Proposition 2.16. Hence, for every finite chain L there is a product S ⊗ L belonging to B. Next, suppose that Age(S) ⊆ B. Let D be a set and L ∶= (D, ≤) be a chain. Let I ′ ⊆ I, V ′ ⊆ V (S), D ′ ⊆ D be three finite sets. Let S ′ ∶= S ↾I ′ ↾V ′ and B ′ ∶= B ↾I ′ . Since S ′ ∈ B ′ , we obtain from our first case applied to S ′ and B ′ that there exists some product

S ↾I ′ ↾V ′ ⊗ L ↾D ′ ∈ B ↾I ′ . By compactness, there is some product R ∶= S ⊗ L on D × V (S) such that Age(R) ⊆ B. Since B is level-finite, B = B and the collection of R ′ such that Age(R ′ ) ⊆ B is reduct-closed by Proposition 2.16, hence Age(R) ⊆ B.
Remark 5.25. If B is not level-finite, the conclusion of the proposition above may be false. Let R ∶= (V, (ρ n ) n∈N ) where V is the set 2 [N] of 0 -1 sequence which are 0 almost everywhere, and where ρ n is the binary relation defined by

ρ n (u, v) ∶= 1 if n is the least integer such that u ↾[n→) = v ↾[n→) and ρ n (u, v) ∶= 0 otherwise. Let B = Age(R).
Then B is inexhaustible and contains no monomorphic structure on three elements. To see the inexhaustibility of B, let t ∈ V and T t ∶ V → V defined by setting T t (u) ∶= u ⊕ t (the Boolean sum of u and t). Then ρ n (u, v) = ρ n (u ⊕ t, v ⊕ t)), hence T u is an automorphism of R. For every n ∈ N, δ n (defined by δ n (m) = 1 if m = n and 0 otherwise), and V n ∶= {u ∈ V ∶ u(m) = 0 for all m ≥ n}, the map T δn maps R ↾Vn to a disjoint copy. The non monomorphy of a 3-element restriction of R is due to the fact that the set of u ↾[n→) for u ∈ V and n ∈ N has the structure of a binary tree; hence, if u, v, w are three distinct elements there is some n such that ρ n (u, v) = ρ n (v, w) = ρ n (u, w) does not hold.

Corollary 5.26. (Corollary IV.3.3.6 of [START_REF] Pouzet | Relation minimale pour son âge[END_REF])Let R be a relational structure and L be a chain. If Age(R) is inexhaustible and level-finite then some product of R and L has the same age as R.

5.4.2.

Almost multichainability and cellularity. We first extend the notion of multichainability to multichainability over a set.

Let R ∶= (V, (ρ i ) i∈I ) be a relational structure. If K is a subset of V , we say that R is multichainable over K, or K-multichainable if there there is an enumeration (a x,y ) (x,y)∈L×W of the elements of V ∖ K by a set L × W , where W is finite and L is a linearly ordered set, such that for every local isomorphism f of L the map (f, 1 W ) extended by the identity on K is a local isomorphism of R (the map (f, 1 W ) is defined by (f, 1 W )(a x,y ) ∶= a f (x),y ). We say that R is almost multichainable if R is multichainable over some finite set F . The pair (|F |, |W |) is the parameter of almost multichainability.

If in the above definition we suppose that L is not ordered and we impose that for every permutation f of L the map (f, 1 W ) extended by the identity on F is a local isomorphism of R then we say that R is cellular.

The notion of almost multichainability was introduced in [START_REF] Pouzet | Sur la théorie des relations[END_REF] (see [START_REF] Pouzet | When is the orbit algebra of a group an integral domain? Proof of a conjecture of P. J. Cameron[END_REF] for further references and discussions). The special case |K| = 1 is the notion of almost chainability introduced by Fraïssé. The notion of cellularity was introduced by Schmerl [START_REF] Schmerl | Coinductive ℵ0-categorical theories[END_REF]. It is illustrated in Section 6 with Theorem 6.17.

Fraïssé's lemma about the almost chainable restriction extends as follows.

Lemma 5.27. Let R be a relational structure with finite signature and domain V , and let K be a finite subset of V . Let S ∈ Age(R). If there are infinitely many restrictions of R isomorphic to S and whose domains are pairwise disjoint, then we may extract a family of these restrictions whose union with K forms a multiple of S by an infinite chain that is K-multichainable.

Proof. The proof is a straightforward application of Ramsey's theorem. Let (F p ) p∈N be a family of pairwise disjoint subsets of V ∖ K such that each restriction R ↾Fp is isomorphic to S by some isomorphism f p from R ↾Fp to S.

Let W be the domain of S, and let m be the maximum of the signature of R. Say that two m-element subsets H and H ′ of N ordered as the integers are equivalent if for the unique order isomorphism h ∶ H ↦ H ′ , the map

⋃ p∈H f -1 h(p) ○ f p from ⋃ p∈H F p to ⋃ p ′ ∈H ′ F p ′
extends the identity on K to a local isomorphism of R. The signature of R being finite, the number of equivalence classes is finite. Hence, from Ramsey's theorem there is an infinite subset X of N such that all m-element subsets belong to the same class. The restriction of R to K ∪ ⋃ p∈X F p has the required property.

Proposition 5.28. Let R be a relational structure with finite signature and domain V , and K ∶= ker(R). Let L ∶= (D, ≤) be a chain. Then there is an extension R ′ on a set V ′ containing V such that Age(R ′ ) = Age(R) and R ′ ↾V ′ ∖K is a multiple of R ↾V ∖K over K by a chain L. Proof. This is a direct consequence of Lemma 5.4.2 above and the Compactness Theorem of first order Logic. This latter result, being applied through the Diagram Method of Robinson as in the proof of Lemma 5.15. Set V ′ ∶= K ∪ (D × (V ∖ K)). To obtain an R ′ satisfying the conclusion of Proposition , give names to the elements of V ′ ; for each element a ∈ D use the natural bijection of V onto K ∪ ({a} × (V ∖ K)) to write axioms expressing that the diagram of R and R ′ ↾K∪({a}×(V ∖K)) are the same. Then express the fact that R ′ ↾V ′ ∖K is a multiple of R ↾V ∖K over K by the chain L. Use Lemma to get the consistency of these axioms. The Compactness Theorem of first order Logic yields the existence of the desired R ′ .

As it stands, the notion of almost multichainability is not flexible enough; in particular, it is not preserved by restriction. For example, the relational structure Q ∶= (Q, ≤, U ), made of the natural order on the rational numbers and a unary relation U dividing Q into two dense sets, is not almost multichainable. But it has the same age as a multichainable relational structure. In fact, a restriction of Q to a subchain of type ω on which the values of U alternate is multichainable with the same age as Q.

In order to clarify this fact (and to rectify Proposition 4.18 of [START_REF] Pouzet | The profile of relations[END_REF]), we find it convenient to introduce the notion of almost multichain -. For that, we recall that a weak order is an order which is the lexicographical sum of antichains indexed by a chain (alternatively, the binary relation x is incomparable to y or x = y is an equivalence relation).

Definition 5.29. An almost multichain -is a relational structure M ∶= (V, ≤, U 1 , . . . , U k , a 1 , . . . a ℓ ), where ≤ is a weak order on V , a 1 , . . . a ℓ are constants, the set F ∶= {a 1 , . . . a ℓ } is an equivalence class attached to ≤ which is before all others, the relations U 1 , . . . U k are unary relations inducing a partition of V ∖ F into chains of the weak order. A relational structure is almost multichainable -if it is freely interpretable by some almost multichain -.

Example 5.30. If R ∶= (V, (ρ i ) i∈I ) is almost multichainable with a set F and an enumeration (a x,y ) (x,y)∈L×W of the elements of V ∖ F by a set L × W , where W is finite and L is a linearly ordered set, then V is the lexicographical sum of F and of the sets W y ∶= {a x,y ∶ x ∈ W } where y ∈ L, ordered according to the chain L. Since the sets U x ∶= {a x,y ∶ y ∈ L} for x ∈ W form a partition of V ∖ L into chains, the corresponding relational structure M is an almost multichain -and interprets freely R.

Other examples are relational structures made of a chain and finitely many unary relations. But, note that not every almost multichainable relational structure has this form. For a simple example, let R be the weak order made of the lexicographical sum of copies of the two element antichain by an infinite chain. This relation is almost multichainable but not freely interpretable by a chain with unary relations added. 5.4.3. Alternative definition. Almost multichainable -structures are freely interpretable by structures made of a weak order whose antichains have finite bounded size and finitely many unary relations. It is not difficult to see that the converse holds. Instead of this notion, we chose a definition ythat is closer to that of almost multichainable structure. Note that with this later definition, no local isomorphism can move an element within an equivalence class. These structures can be described as follows:

Proposition 5.31. The class of almost multichainable -relational structures is the least class M of relational structures containing each weak order whose antichains have a finite bounded size, containing every relational structure freely interpretable by some member of M and containing every member of M with some unary relation added.

From the definition of almost multichain -, the restriction of an almost mutichain -is an almost multichain -. Applying the Compactness Theorem of first order logic in the form of (2) of Lemma 2.18 (with R replaced by M and S replaced by R), we get that if R is an almost multichainable -structure and the age of some R ′ is included into the age of R then R ′ is almost multichainable -. This does not say how almost multichainable -structures relate to those that are almost multichainable. The answer is the following, but it requires a proof. Theorem 5.32. An infinite relational structure R is almost multichainable -if and only if some restriction of R is almost multichainable and has the same age as R.

In particular, infinite ages of almost multichainable -structures and ages of almost multichainable structures coincide. Before proving this result, we obtain the following consequence.

Corollary 5.33. If an infinite age A is included in the age of an almost multichainable structure, then A is the age of an almost multichainable structure.

Proof. Let B be the age of an almost multichainable structure R containing A. Then R is almost multichainable -. According to Theorem 5.32 some restriction of R is almost multichainable and has the same age as R. This proves our claim.

The proof of Theorem 5.32 is based on the notion of wqo and Higman's theorem. We prove first: Theorem 5.34. The age of an almost multichainable relational structure, and more generally of an almost multichainable -relational structure, is wqo. In fact, it is hereditarily bqo.

Proof. Part 1 of Theorem 4.19 [START_REF] Pouzet | The profile of relations[END_REF] asserts that the age of an almost multichainable relational structure is very-well-quasi-ordered. The proof consisted in encoding members of the age by words over a finite alphabet and apply Higman's Theorem on words. The proof applies with no changes to almost multichainable - relational structures. In fact, it only relies on the fact that the age of a chain labelled by a wqo is wqo. Indeed, we claim that this property ensures that the age of a structure M ∶= (V, ≤, U 1 , . . . U k ) made of a weak order whose antichains have finite bounded size and with finitely many unary relations U 1 , . . . , U k is wqo. Since the age of an almost multichainable -relational structure is freely interpretable by some M of the form above, it is wqo too. To deduce our claim, we prove that members of the age of M can be encoded by labelled chains (as we have seen, they are not necessarily interpreted by labelled chains). For that, let L be the chain of equivalence classes of the weak order ≤. Then, write V as the lexicographical sum of the equivalence classes K indexed by L. To each finite subset H of V associate the sequence K 1 , . . . K n of the equivalence classes on which K i ∩ H is nonempty, this sequence being enumerated in the order of L. Then, to this sequence associate the word C H ∶= H K 1 ⋯H Kn whose letters H K i are the sets H K i ∶= {j ∶ U j ∩ K i ∩ H ≠ ∅}. Order the letters by set inclusion. From Higman's theorem, the collection of these words forms a wqo and in fact a bqo. The encoding of M ↾H 's by the C H 's is such that

C H ≤ C H ′ implies that M ↾H ≤ M ↾H ′ .
The fact that Age(M ) is bqo follows. Adding a coloring of V by unary relations or by members of a wqo will not change the outcome. Hence Age(M ) is hereditarily bqo.

Due to the encoding of the members of an age of an almost multichainable structure by words over a finite alphabet, we have the following result (see part 2 of Theorem 4.19 of [START_REF] Pouzet | The profile of relations[END_REF]).

Theorem 5.35. The height of the age of almost multichainable structures is always strictly less than ω ω .

Proof. Let R be an almost multichainable structure. Then R is freely interpretable by a relational structure M as in Example 5.30. Let (k, p) be the parameters of M . We claim that (5.3) H(Age(R)) ≤ H(Age(M )) < ω 2 p -1 + ω.

The first inequality relies on a general property of free interpretability.

Lemma 5.36. If the set Id(Age(M )), ordered by set inclusion, of ages included in the age of a relational structure M is well-founded, then for every relational structure R, freely interpretable by M , the set Id(Age(R)) is well founded and

(5.4) H(Age(R)) ≤ H(Age(M )).

Proof. Let R be freely interpretable by M , and let P be a free operator transforming M into R. The latter induces a transformation of the subages of M into subages of R. According to (2) of Lemma 2.18, for each age A belonging to Id(Age(R)) there is some age M belonging to Id(Age(M )) such that P transforms M to A. This implies that Id(Age(R)) is well founded as soon as Id(Age(M )) is. The inequality follows from a straightforward induction.

For the second inequality, let (k, p) be the parameters of M . Suppose first that k = 0, that is, ker(M ) is empty. In this case, Age(M ) is essentially the free ordered monoid generated by the collection of nonempty subsets of {1, . . . , p} ordered by set inclusion. If we order these subsets by the equality relation, the free ordered monoid is Age(Q, U 1 , . . . , U 2 p -1 ) and hence, o(Age(Q, U 1 , . . . , U 2 p -1 )) = ω ω 2 p -2 from Example 3.16. We have o(Age(M )) ≤ o(Age(Q, U 1 , . . . , U 2 p -1 )), and thus o(Age(M )) ≤ ω ω 2 p -2 .

From Theorem 3.15, we have o(Age(M )) = ω α ⋅ q with ω ⋅ α ≤ H(Age(M )) < ω ⋅ (α + 1). Hence, α ≤ ω 2 p -2 and thus H(Age(M )) < ω ⋅ (ω 2 p -2 + 1) = ω 2 p -1 + ω. If k / = 0, then according to (2) of Lemma 3.34, H(Age(M )) = H(Age(M ′ ))+r, where M ′ is a reduction over the kernel of M and r < ω. This yields H(Age(M )) < ω 2 p -1 + ω as claimed. Note that for p = 1, R is almost chainable and so H(Age(R)) < ω ⋅ 2, as already seen. 5.4.4. Towards a proof of Theorem 5.32. The proof uses the notion of indecomposable labelled chain and well-quasi ordering. A labelled chain is a pair (C, ℓ) where C is a chain and ℓ is a map from C into a poset P . Chains labelled by the same poset can be compared by embeddability, like one compares chains with a fixed finite number of unary relations added. We say that a labelled chain (C, ℓ) is age-indecomposable if for every partition of the chain C into an initial and a final interval I and J, the age of one of the restrictions (C, ℓ) ↾I (C, ℓ) ↾J is equal to the age of (C, ℓ). Lemma 5.37. If a chain (C, ℓ) is labelled by a finite poset or more generally by a wqo then C decomposes into finitely many disjoint intervals I 1 , . . . , I i , . . . I n such that each restriction (C, ℓ) ↾I i is ageindecomposable.

Proof. The proof is typical of the strength of the notion of wqo. Suppose that there is some labelled chain for which the above property does not hold. Since the age of every chain labelled by a wqo is wqo, the collections of these ages is well-founded w.r.t. set inclusion. Hence, there is a labelled chain (C, ℓ) for which the above property does not hold and the age of (C, ℓ) is minimal. We claim that (C, ℓ) is ageindecomposable. Indeed, let if for every partition I, J of the chain C into an initial and a final interval, the ages of the restrictions (C, ℓ) ↾I and (C, ℓ) ↾J are smaller that the age of (C, ℓ) then, due to the minimality of that age, each of these restriction has a finite decomposition, hence (C, ℓ) has a finite decomposition. Since we supppose it has not such a decomposition, one of these ages is the same as the age of (C, ℓ). But then (C, ℓ) is age-indecomposable, contradicting our hypothesis. Lemma 5.38. Every infinite and age-indecomposable labelled chain (C, ℓ), labelled by a finite poset, embeds a labelled chain with the same age as (C, ℓ) which is an ω or ω * -sum of copies of a finite labelled chain (Q, q). Proof. Let (q 1 , . . . , q m ) be an enumeration of the set of maximal elements of the range of ℓ. We may view this enumeration as a labelled chain (Q, q). Since (C, ℓ) is age-indecomposable, (Q, q) embeds in (C, ℓ). From Ramsey's theorem, an ω-sum or an ω * -sum of copies of (Q, ℓ) embeds in (C, ℓ). It has the same age as (C, ℓ).

Proof of Theorem 5.32. Let R be an almost multichainable -relational structure. Let M ∶= (V (R), ≤, U 1 , . . . , U k , a 1 , . . . a ℓ ) be an almost multichain -interpreting R. Let L be the chain of equivalence classes K of the weak order ≤. For each K set ℓ(K) ∶= {j ∶ U j ∩ K ≠ ∅}. Looking at these sets ℓ(K) as letters, these letters being subsets of {1, . . . , k} ordered by set inclusion, we get the labelled chain (L, ℓ). Applying Lemmas 5.37 and 5.38 we get that (L, ℓ) embeds a finite lexicographical sum (L , ℓ 1 ) + ⋅ ⋅ ⋅ + (L , ℓ m ) where (L i , ℓ i ) is either finite or an ω or an ω * -sum of copies of a finite labelled chain (Q i , q i ). Let I be the set of infinite L i 's. For each i ∈ I, L i decomposes into an union

⋃ n∈ω Q i,n such that (Q i , ℓ ↾Q i,n ) is isomorphic to (Q i , q i ). For n ≤ ω, let K n ∶= ⋃ 1≤i≤m Q i,n and let F ∶= ⋃ i/ ∈I L i .
Let V ′ be the set union of F and the K n 's and let ≤ ′ be the weak order on V ′ whose classes are F and the K n 's ordered as ω. Adding to the restrictions of the U j to V ′ some unary relations so that no element on an equivalence class can be moved by a local isomorphism we get an almost multichain. It Proof. Let L be an infinite chain. According to Corollary 5.26 some product R ⊗ L has the same age as R.

Let (V n ) n∈N be an increasing sequence of finite subsets V n of V whose union is V (R). Set

A n ∶= Age((R ⊗ L) ↾ (V (L) × V n )).
Suppose that A has a height. Since A n is inexhaustible H(A n ) is a limit ordinal (Corollary 3.38). Since H(A) < ω 2 , the sequence (A n ) n∈N must be stationary. So Age(A) = A n for some n ∈ N.

Proposition This result is stated as Théorème III-1.1. in "Caractérisation combinatoire et topologique des âges les plus simples" [START_REF] Pouzet | Sur la théorie des relations[END_REF]. This is a nontrivial result. It is an immediate consequence of Proposition 5.41 and of the following result. Proposition 5.43. Let A be the age of a relational structure with finite signature. If H(A) < ω 2 , then A is almost inexhaustible.

A proof of this proposition was given in [START_REF] Pouzet | The profile of relations[END_REF] Theorem 4.24, p. 267, for binary structures, and also in Proposition 4.6 of this paper. A proof of an equivalent statement is given in "II-Théorème central" of "Caractérisation combinatoire et topologique des âges les plus simples" [START_REF] Pouzet | Sur la théorie des relations[END_REF]. This statement is the following: Proposition 5.44. Let A be the age of a relational structure with finite signature. If every proper subage of A is very-well-quasi-ordered, then either A is almost inexhaustible or H(A) ≥ ω 2 .

Let us prove that Proposition 5.43 and Proposition 5.44 are equivalent. Suppose Proposition 5.43 true. We derive Proposition 5.44 as follows. Let A be the age of a relational structure with finite signature such that every proper subage of A is very-well-quasi-ordered. Necessarily, Id(A) is well-founded, hence H(A) is well defined. If A is not almost inexhaustible, then H(A) ≥ ω 2 by Proposition 5.43. Hence A satisfies the conclusion of Proposition 5.44. Conversely, suppose Proposition 5.44 true. Let A be the age of a relational structure with finite signature. If A is very-well-quasi-ordered, it is 1 --wqo, hence almost inexhaustible by Corollary 4.5. If A is not very-well-quasi-ordered but H(A) < ω 2 , then A contains a minimal subage A ′ which is not very-well-quasi-ordered. Since A ′ ⊆ A, H(A ′ ) < H(A) < ω 2 . Proposition 5.44 ensures that A ′ is almost inexhaustible. By Proposition 5.41, A ′ is the age of a almost multichainable structure. By Theorem 5.34 it is hereditarily wqo, which is a contradiction.

The proof of Proposition 5.44 has two parts. First, we show in Proposition 5.49 below, that the collection of subages of A contains chains of type ω ⋅ n for every integer n provided that the complement of the kernel ker(R) of some R with age A is not val-finite, a notion invented by Fraïssé [START_REF] Fraïssé | L'intervalle en théorie des relations, ses généralisations, filtre intervallaires et clôture d'une relation[END_REF], Section 4, p.325, that we will recall below in Subsection 5.5.1. Supposing that every proper subage of A is hereditarily wqo, Id(A) is well-founded. Under the hypothesis on R, H(A) ≥ ω 2 .

Next, we suppose that for every R with age A, the complement of ker R is val-finite. Then we show that R is freely interpretable by some relational structure M such that ker(R) and V ∖ ker(R) are interval of M and the restrictions M ↾V ∖{x} for x ∈ ker(R) are hereditarily wqo. We show then that this is impossible, completing the proof of Proposition 5.44.

5.5.1.

Val-finite subsets of a relational structure and wqo. The notion of fini-val (in french), invented by Fraïssé, generalizes the notion of module also introduced by him under the name of interval and independently by Gallai [START_REF] Gallai | Transitiv orientbare graphen[END_REF].

Let R be a relational structure with domain V , K be a subset of V and m be an nonnegative integer. Two m-tuples x ∶= (x 1 , . . . , x m ) and x ′ ∶= (x ′ 1 , . . . , x ′ m ) belonging to (V ∖ K) m are m-equivalent over K when the transformation of x into x ′ extended by the identity on K induces a local isomorphism of R. This defines an equivalence relation on (V ∖K) m . We denote by θ R,K (m) the number of these equivalence classes. Then the subset V ∖ K is said to be val-finite over K if for each positive integer m the number θ R,K (m) is finite. Recall that a subset A of the domain V of a relational structure R is an interval if every local isomorphism of R ↾A extended by the identity on V ∖ A is a local isomorphism of R. For example, if R is made of binary relations, each one being either reflexive or irreflexive, then A is an interval if and only if it is autonomous (as defined in Subsection 3.3.3). As observed by Fraïssé, one has: Lemma 5.45. Let R be a relational structure with a finite signature and let A be a subset of its domain V . If A is an interval, then A is val-finite over V ∖ A. Moreover A is val-finite over V ∖ A if and only if there is some relational structure M with finite signature interpreting R and such that A is an interval of M .

For the proof of the second assertion, note that if m is the maximum of the signature of R, then since the number θ R,A (m) of equivalence classes over V ∖ A is finite, we may find a finite subset F ⊆ V ∖ A such that two m-uples of A that are equivalent over F are equivalent over V ∖ A. If we label the elements of F with constants a 1 , . . . , a k (k ∶= |F |), then the relational structure M ∶= R ⋅ a 1 ⋅ ⋯ ⋅ a k has the required property. From this proof, we may extract the following characterization. Lemma 5.46. Let R be a relational structure with domain V and a finite signature and let A be a subset of V . Then A is a val-finite subset of V if and only if there is some finite subset F of V ∖ A such that the local isomorphisms of R ↾A , which can be extended by the identity on V ∖ A to local isomorphisms of R, are the proof of Proposition 5.49.

To complete the proof of Theorem 5.42 we will need the following result. Proposition 5.50. Let R with domain V and let A be and infinite subset of V such that V ∖ A is val-finite, and for each x ∈ L, Age(R ↾ {x}) is very-well-quasi-ordered. Then the following assertions hold.

(1) R is freely interpretable by some relational structure M for which A and V ∖ A are two intervals, and the age of the restriction M ↾ V ∖ {x} is very-well-quasi-ordered for every x ∈ V . (2) Suppose that A = ker R, and let k < k ′ be two integers, x 0 ∈ A, and let (M n ) n∈∈N be a sequence of relational structures

M n ∶= (R n , U 1 n , . . . , U k ′ n ) where R n ∶= R ↾ F n is a finite restriction of R, each U i n is a subset of F n for 1 ≤ i < k ′ , with U i n ⊆ U k+1 i for k + 1 ≤ i ≤ k ′ . If for each x ∈ V , the set {n ∈ N∶ x ∈ U k+1
n } is finite, then there are n, m ∈ N, n < m, and an embedding

f ∶ M n ∖ {x 0 } ↦ M m ∖ {x 0 } such that f ↾U k+1 n
extended by the identity on x 0 is an embedding of (2). This sentence is only interesting if all the F n contain x 0 . Supposing that the conclusion does not hold, we derive a contradiction. We may suppose without loss of generality that this sequence is minimal in the Nash-Williams sense. We may go to the "limit", constructing a multirelational structure

M ↾U k+1 n ∪{x 0 } into M ↾U k ′ n ∪{x 0 } . Proof. (1). Since V ∖ A is val-finite, A
M ω ∶= R ω ⋅ U 1 ω ⋅ ⋅ ⋅ U k ′ ω
, such that for every proper restriction of M n , there is some embedding f satisfying the conditions above. We may chose for R ω a restriction of some extension of R such that all x ∈ K are outside the union of the U k+1 ω , . . . , U k ′ ω 's. By (1), the local automorphisms of M ω that induce local isomorphisms of M ω ↾ {x 0 } ∪ U k+1 ω , are the local isomorphisms of a relational structure whose age is very-well-quasi ordered. Since one can transform the M n 's into bounds of this relational structure, and since there are only finitely many, we get a contradiction. 5.5.2. End of the proof of Theorem 5.42. Let A be the age of an almost multichainable relational structure with H(A) < ω 2 . According to Proposition 5.49, the complement of ker(R) of every R with age A is val-finite over ker(R).

Our aim is to prove that ker(R) is finite. Suppose that this is not the case. Consider R with age A, countable base V , and infinite kernel K. According to the equivalence between Proposition 5.43 and Proposition 5.44, we may suppose that the age of R ↾V ∖{x} is hereditarily wqo, for every x ∈ V . In order to get a contradiction, let us enumerate the members of V into a chain C order isomorphic to the chain N of positive integers, and let us suppose that the first element x 0 of C is in K. Since K is infinite, we may find an infinite sequence of finite subsets F n of V with a n ∈ F n ∩ K such that the sequence of restrictions B n ∶= R ↾Fn ⋅ a n forms an antichain with respect to embedabbility. This sequence being bad, we may suppose that it is minimally bad in the Nash-Williams sense (that is, no sequence with a strictly smaller prefix can be bad). Due to our hypothesis on R, only finitely many F n do not contain x 0 . In fact, due to the minimality, all F n contain x 0 . Now we proceed as in [START_REF] Pouzet | Un belordre d'abritement et ses rapports avec les bornes d'une multirelation[END_REF]. In order make the proof simpler, we suppose that R is made of binary relations (the present proof extends to arbitrary arities, while the one given in (1) of Theorem 4.24, p. 267, [START_REF] Pouzet | The profile of relations[END_REF], does not). To B n we associate the relational structure M ∶= R ↾Fn ⋅ U n ⋅ b n , where U n is a maximal final section of C ↾Fn and b n is the element just before U n such that for infinitely many n ′ there exists f defined on F n and with values in F n ′ , inducing an embedding of B n-x 0 in B n ′ and an embedding of

B n ↾ {x 0 , a n } ∪ U n ⋅ x 0 in B n ′ .
There are not infinitely many integers n such that for any two of them, say, n and n ′ , a map f from F n to F n ′ , which leaves x 0 fixed,

• transforms a n and b n into a n ′ and b n ′ , respectively, • sends U n into U n ′ , and • induces an embedding of B n-x 0 into B n ′ , and an embedding of B n ↾ {x 0 , a n } ∪ U n into B n ′ . According to (2) of Proposition 5.50, we may suppose that there is some element x 1 in K which belongs to infinitely many U n 's. Since the B n 's are incomparable this is impossible.

This completes proof of Theorem 5.42. Problem 17. Describe ages of multichainable relational structures R such that H(Age(R)) < ω 2 . 5.6. Monomorphic decomposition. I present below the notion of monomorphic decomposition of a relational structure introduced in [START_REF] Pouzet | Some relational structures with polynomial growth and their associated algebras I. Quasi-polynomiality of the profile[END_REF] with an approach presented in [START_REF] Oudrar | Sur l'énumération de structures discrètes: une approche par la théorie des relations[END_REF]. A relational structure admitting a finite monomorphic decomposition is a very particular kind of almost multichainable structure. Its age has height strictly smaller than ω 2 . These structures play a pivotal role in the study of age algebras.

Let R be a relational structure. A subset V ′ of V (R) is a monomorphic part of R if for every integer k and every pair A, A ′ of k-element subsets of V (R), the induced structures on A and A ′ are isomorphic whenever A∖V ′ = A ′ ∖V ′ . A monomorphic decomposition of R is a partition P of V (R) into monomorphic parts. A monomorphic part which is maximal for set inclusion among monomorphic parts is a monomorphic component of R. The monomorphic components of R form a monomorphic decomposition of R of which, every monomorphic decomposition of R is a refinement (Proposition 2.12 of [START_REF] Pouzet | Some relational structures with polynomial growth and their associated algebras I. Quasi-polynomiality of the profile[END_REF]). This decomposition is denoted by P(R) and called minimal or canonical and the number of its infinite blocks is called the monomorphic dimension of R. The canonical decomposition can be introduced via an equivalence relation as follows. Let R be a relational structure and V ∶= V (R). Let x and y be two elements of V . Let A be a finite subset of V ∖ {x, y}, we say that x and y are A-equivalent and we set x ≃ A,R y if the restrictions of R to {x} ∪ A and {y} ∪ A are isomorphic. Let k be a nonnegative integer, we set x ≃ k,R y if x ≃ A,R y for every k-element subset A of V (R) ∖ {x, y}. We set x ≃ ≤k,R y if x ≃ k ′ ,R y for every k ′ ≤ k and x ≃ R y if x ≃ k,R y for every k. For example, the ∅-equivalence is an equivalence relation on V , two vertices x and y being equivalent if and only if the restrictions of R to x and y are isomorphic. As it is easy to see: Corollary 5.55. The relations ≃ k,R , ≃ ≤k,R and ≃ R are equivalence relations on V . Furthermore, each equivalence class of ≃ R is a monomorphic part of R, and each monomorphic part is included in some equivalence class.

The latter result yields: Proposition 5.56. The equivalence classes of ≃ R form a decomposition of R into monomorphic parts and every decomposition into monomorphic parts is finer. Thus, the decomposition of R into equivalence classes of ≃ R coincides with the decomposition of R into monomorphic components.

Theorem 5.57. A relational structure R ∶= (V, (ρ i ) i∈I ) admits a finite monomorphic decomposition if and only if there exists a linear order ≤ on V and a finite partition (V x ) x∈X of V into intervals of C ∶= (V, ≤) such that every local isomorphism of C which preserves each interval is a local isomorphism of R. Moreover, there exists such a partition whose number of infinite blocks is the monomorphic dimension of R.

In other words, R ∶= (V, (ρ i ) i∈I ) admits a finite monomorphic decomposition if and only if R is freely interpretable by a structure of the form S ∶= (V, ≤, (u 1 , . . . , u l )) where ≤ is a linear order and u 1 , . . . , u l are finitely many unary relations determining a partition of V into intervals of (V, ≤).

A straightforward application of the Compactness Theorem of First Order Logic yields:

Theorem 5.58. A relational structure R admits a finite monomorphic decomposition if and only if there is some integer ℓ such that every member of Age(R) has a monomorphic decomposition into at most ℓ classes.

The following property allows to test the existence of a finite monomorphic decomposition by looking at the relations composing R, see [START_REF] Oudrar | Sur l'énumération de structures discrètes: une approche par la théorie des relations[END_REF], p. 168 and [START_REF] Oudrar | Ordered structures with no finite monomorphic decomposition[END_REF].

Proposition 5.59. Let m ∈ N. An ordered relational structure R ∶= (V, ≤, (ρ i ) i<m ) has a finite monomorphic decomposition if and only if every R i ∶= (V, ≤, ρ i ) (i < m) has such a decomposition.

WQO AGES AND PROFILE

As already said, the profile of a hereditary class C of finite relational structures is the function φ C which counts for every integer n the number φ C (n) of members of C on n elements, isomorphic members being identified. We say that the profile of a relational structure R, denoted by φ R , is the profile of its age. Hence, φ R (n) is the number of induced substructures of R on n elements, isomorphic substructures being identified. There is a parent notion, the labelled profile; instead of counting up to isomorphism, one counts all possible structures on n elements. Since n-element chains have no nontrivial automorphism, the labelled profile of a hereditary class C is the profile of the class C.L made of structures of the form (S, L) where S ∈ C and L is a linear order on the same domain as S. This notion started with Scheinerman and Zito [START_REF] Scheinerman | On the size of hereditary classes of graphs[END_REF]. The two notions have a common feature. There are jumps in their behavior. Several dozen papers have been written on these notions. We will not attempt to present a synthesis of results but instead will present results, which have interested me since Fraïssé and I started to think about the profile (the name of "profile" for this counting function was proposed to us by William Craig in 1969). We will present an overview, with results of Cameron on the orbital profile in that setting, and conclude with a recent result by Falque and Thiéry [START_REF] Falque | Macpherson's conjecture holds: the orbit algebra of a permutation group with polynomial profile is finitely generated[END_REF] solving conjectures of Cameron and Macpherson.

6.1. A basic result. Theorem 6.1. The profile of an infinite relational structure is non-decreasing.

The result was conjectured with Fraïssé in [79] (Problème 2.5.3, p.1627). The proof, based on Ramsey's theorem, appears in Fraïssé's Cours de Logique [START_REF] Fraïssé | Cours de Logique Mathématique, tome 1, Relation et formule logique[END_REF] in 1971 (Exercice 8, p.113). The result was improved in 1976 by showing that φ R (n) ≤ φ R (n + p) provided that the domain of R, possibly finite, has at least 2n + p elements (with n + p ≥ n ≥ 0) [START_REF] Pouzet | Application d'une propriété combinatoire des parties d'un ensemble aux groupes et aux relations[END_REF]. The proof of this improvement is based on the non-degeneracy of the Kneser matrix of the n-element subsets of a 2n + p element set, a result obtained independently by Gottlieb [START_REF] Gottlieb | A class of incidence matrices[END_REF] and Kantor [START_REF] Kantor | On incidence matrices of finite projective and affine spaces[END_REF].

With what we did in Subsection 5.3, the proof of Theorem 6.1 is almost immediate. Indeed, let R ∶= (V, (ρ i ) i∈I ) be an infinite relational structure and n be an integer. Then, by a double counting argument, one gets φ R (n) ≤ (n + 1)φ R (n + 1). In particular, if φ R (n) is infinite then φ R (n + 1) is infinite too and φ R (n) ≤ φ R (n+1). Thus, one may suppose φ R (n) finite. In this case, there is a finite subset

I ′ of I such that the reduct R ′ ∶= R ↾I ′ satisfies φ R ′ (n) = φ R (n)
and there is a finite subset K of V containing a representative of each isomorphism type of n-element restrictions of R ′ . According to Lemma 5.14, there is an infinite subset

V ′ of V containing K on which the restriction R ′′ ∶= R ′ ↾V ′ is K-chainable. According to Theorem 5.13, φ R ′′ (n) ≤ φ R ′′ (n + 1). Since φ R (n) = φ R ′ (n) = φ R ′′ (n) and φ R ′′ (n + 1) ≤ φ R ′ (n + 1) ≤ φ R (n + 1), the inequality φ R (n) ≤ φ R (n + 1) follows.
This result points to the importance of the notion of an almost chainable structure. It suggests to first look at hereditary classes with bounded profile and beyond. We do that in the next subsection. 6.2. Bounded profile and beyond. The notion of profile leads to complete Theorem 5.22 with the following characterization (see [START_REF] Fraïssé | Interprétabilité d'une relation par une chaîne[END_REF], 2.5.2, p. 1627):

Theorem 6.2. The profile φ R of a relational structure R is bounded above, that is, Age(R) is finimorphic, if and only if R is almost chainable.
With what we know, the proof is immediate: if R is almost chainable then φ R is bounded above by 2 | ker(R)| where ker(R) is the kernel of R. For the converse, the conclusion is immediate if R is finite. If R is infinite then since φ R is bounded above, Age(R) is level-finite. Hence (2) of Proposition 5.18 applies: Age(R) is an increasing union of a sequence of infinite almost chainable ages. Since φ R is bounded, this sequence is stationary. The conclusion follows.

Let G be a permutation group acting on a set V and let R be a relational structure encoding G. Then, the kernel of R is the union U of the finite G-orbits of the one-element sets (hence the kernel of R is empty if G has no finite orbit).

Closed monomorphic groups are described in Theorem 5.9. In terms of orbital profile these groups are those of orbital profile equal to 1.

From Theorem 6.2, we immediately have this result.

Theorem 6.3. The orbital profile of a permutation group G acting on a countable set V is bounded if and only if U , the union of all finite orbits of G, is finite and G acts on V ∖ U as a monomorphic group.

The following result illustrates Theorem 5.23 (the crucial part is the implication (iv) ⇒ (ii) of Theorem 5.23). Theorem 6.4. The profile φ C of a hereditary class C of relational structures is bounded above, that is, C is finimorphic, whenever C is a finite union of ages of almost chainable structures. The converse holds if the signature is finite.

As already seen, without a condition like the finiteness of the signature, there are hereditary classes of relational structures whose profile is bounded above and which are not finite unions of ages of almost chainable structures.

Above bounded profile, we have: Lemma 6.

≤ ψ(n) ≤ n + 1 for all n ∈ N. For A ∶= {n ∈ N ∶ ψ(n) < ψ(n + 1)}, we have φ R A ≤ ψ. Indeed, let n ∈ N and {n 1 < ⋅ ⋅ ⋅ < n k } = A ∩ (← n[. From 1 ≤ ψ(n 1 ) < ψ(n 1 + 1) < ⋅ ⋅ ⋅ < ψ(n k ) < ψ(n k + 1) ≤ ψ(n), we get ψ(n) ≥ k + 1 = φ R A (n).
Finally, let φ ∶ N → N be any non-decreasing and unbounded map; we may select ψ ∶ N → N with ψ non-decreasing and unbounded and eventually dominated by φ such that

1 ≤ ψ(n) ≤ n + 1 for all n. Then, for n large, φ R A (n) ≤ ψ(n) ≤ φ(n).
The reader will notice that in these examples, if φ is unbounded then the signature of R A is unbounded and also the kernel of R A is infinite (equal to N). Let us see what happens if the signature or the kernel is finite.

6.3. Jumps in the profile. At the end of the 1970's, I noted that beyond bounded profiles, and provided that the relational structures satisfy some mild conditions, there are jumps in the behavior of their profiles, for instance, no profile grows as log n or n log n.

In 1981, I proved that if the profile φ C of a hereditary class C is unbounded then its growth is at least linear (in the sense that

φ C (n) n
≥ a for some constant a > 0 and all n) provided that the signature is bounded [START_REF] Pouzet | Application de la notion de relation presque-enchaînable au dénombrement des restrictions finies d'une relation[END_REF] . Thus is due to the fact that if m is the maximum of the arity of the signature µ there is some integer s(m) such that if the profile φ C is constant from n to n + s(m) + 1 then it remains constant afterwards. The existence of s(m) relies on Frasnay's theory.

Cameron [START_REF] Cameron | Orbits of permutation groups on unordered sets[END_REF] proved a similar result for the orbital profile of groups (see Subsection 1.4.6). If the orbital profile ϑ G of a group G is unbounded then ϑ G (n) ≥ n 2 -c for some constant c and all n. Let us say that the growth of φ R is polynomial of degree

k if a ≤ φ R (n) n k
≤ b where a and b are positive constants; the growth is faster than every polynomial if φ R (n) n k goes to infinity with n for every k. In my thesis [START_REF] Pouzet | Sur la théorie des relations[END_REF], I obtained the following with a different approach. Theorem 6.6. The growth of the profile φ R of a relational structure R ∶= (V, (ρ i ) i∈I ) is either polynomial or faster than every polynomial provided that either the signature µ ∶= (m i ) i∈I is bounded above or the kernel ker(R) of R is finite.

The fact that the signature µ ∶= (m i ) i∈I is bounded in Theorem 6.6 amounts to the fact that R and some finite reduct have the same profile. This is essentially Lemma 2.8 of [START_REF] Pouzet | Application de la notion de relation presque-enchaînable au dénombrement des restrictions finies d'une relation[END_REF]. For the reader's convenience, we repeat the argument. Lemma 6.7. Let R ∶= (V, (ρ i ) i∈I ) be a relational structure with signature µ ∶= (m i ) i∈I . If µ is bounded and φ R (n) is finite for some n ≥ max µ then there is a reduct R ↾I ′ of R with I ′ finite and with the same profile as R. In particular φ R (n ′ ) is finite for every integer n ′ .

Proof. Let q ∶= φ R (n) and F 1 , . . . , F q be q subsets of V such that each n-element restriction of R is isomorphic to one of the restrictions R ↾F 1 , . . . , R ↾Fq . Let Q ∶= F 1 ∪ ⋅ ⋅ ⋅ ∪ F q . Say that two indices i, i ′ ∈ I are equivalent if m i = m i ′ and the restrictions of ρ i and ρ i ′ to Q are equal. We claim that i and i ′ are equivalent if and only if ρ i = ρ i ′ . Indeed, suppose that i and i ′ are equivalent. Then m i = m i ′ . Let m be the common value. Let (x 1 , . . . , x m ) ∈ V m , we prove that ρ i (x 1 , . . . , x m ) = ρ i ′ (x 1 , . . . , x m ). For that, extend {x 1 , . . . x m } to a n-element subset F of V . Since the restriction R ↾F is isomorphic to some R ↾F q ′ , let ψ such an isomorphism. Since i and i ′ are equivalent, ρ i (ψ(x 1 ), . . . , ψ(x m )) = ρ i ′ (ψ(x 1 ), . . . , ψ(x m )). Since ψ is an isomorphism ρ i (x 1 , . . . , x m ) = ρ i (ψ(x 1 ), . . . , ψ(x m )) and also ρ i ′ (x 1 , . . . , x m ) = ρ i ′ (ψ(x 1 ), . . . , ψ(x m )). Hence ρ i (x 1 , . . . , x m ) = ρ i ′ (x 1 , . . . , x m ) as claimed. The equivalence relation defined above has finitely many classes. Taking a representative in each class yields a finite set I ′ . The reduct R ↾I ′ has the same local isomorphisms as R, hence the same profile.

In order to prove Theorem 6.6, we relate the growth of the profile of a relational structure R to the height H(Age(R)) of the set Id(Age(R)) of subages of Age(R). We prove: Theorem 6.8. Let R be a relational structure such that the signature µ ∶= (m i ) i∈I is bounded above or the kernel ker(R) of R is finite. Then, either R is almost multichainable and H(Age(R)) = ω ⋅ (k + 1) + p with p < ω, in which case φ R grows as a polynomial of degree k, or φ R grows faster than every polynomial.

A proof, complete for binary structures, is given in [START_REF] Pouzet | The profile of relations[END_REF], Theorem 4.30, p. 269. Here are the essential ingredients of the proof and their organisation. 6.3.1. Ingredients for the proof of Theorem 6.8. First, in order to prove that the growth of the profile of a relational structure R with finite kernel is polynomial, we show that it suffices to prove that the growth of a reduction over its kernel is polynomial.

For that, we say that the growth of a map φ ∶ N → N is invariant under translation if φ and its translate n → φ(n + 1) have the same growth. In this case, φ is bounded by an exponential function. Item (3) of Lemma 3.34 asserts that the growth of φ R is invariant under translation if and only if the growth of reduction φ M is invariant too, in which case there are the same. Since polynomial functions are invariant under translation, we get: Lemma 6.9. If ker(R) is finite then φ R is bounded by a polynomial of degree k, resp. has polynomial growth of degree k, if and only if the profile φ M of a reduction M is bounded by a polynomial of degree k, resp. has polynomial growth of degree k.

Next, we use Proposition 5.41, Theorem 5.42 and the following two statements: Theorem 6.10. If the age Age(R) of a relational structure R is inexhaustible and

H(Age(R)) ≤ ω ⋅ (k + 1), then φ R (n) ≤ ( n + k k ),
for every integer n.

Lemma 6.11. If the age Age(R) of a relational structure R is inexhaustible and H(Age(R)) = ω ⋅ (k + 1), then φ R grows as a polynomial of degree k.

Theorem 5.42 and these two statements are proved in [START_REF] Pouzet | The profile of relations[END_REF] (see Theorem 4.28, Lemma 4.29, p. 269, and (1) of Theorem 4.24) except that the proof of Theorem 5.42 ((1) of Theorem 4.24) is given only for binary structures. 6.3.2. Structure of the proof of Theorem 6.8. Let R be a relational structure such that the signature µ ∶= (m i ) i∈I is bounded above or the kernel ker(R) of R is finite. If for some integer n, φ R (n) is infinite then φ R (n ′ ) is infinite for all n ′ ≥ n hence the growth of φ R is faster than every polynomial.

Case 1. Assume that ker(R) is finite, that is, R is almost inexhaustible. In this case, a reduct M of R is inexhaustible. According to Proposition 5.41, either M is multichainable of height ω (k + 1) or Age(M ) contains multichainable ages of height ω ⋅ (k + 1), for every integer k. Subcase 1. H(Age(M )) = ω ⋅ (k + 1). Then φ M grows as a polynomial of degree k from Theorem 6.10 and Lemma 6.11. According to Lemma 3.34, H(Age(R)) = ω ⋅ (k + 1) + p. Lemma 6.9 asserts that φ R grows as a polynomial of degree k. Subcase 2. Age(M ) contains multichainable ages of height at least ω ⋅ (k + 1) for arbitrarily large finite k. Since by Subcase 1 their profiles is polynomial of degree at least k, the profile of Age(M ) grows faster than every polynomial. Lemma 6.9 asserts that the same holds for φ R .

Case 2. Assume that ker(R) is infinite. Hence, from our hypothesis, the signature µ is bounded. In this case, Age(R) contains a subage of height ω 2 . Indeed, either Age(R) is wqo, hence it has a height and by Theorem 5.42 this height is at least ω 2 , or Age(R) is not wqo and, by Theorem 5.51, it contains a subage of height ω 2 . According to Theorem 5.42 the proper subages are almost multichainable and according to Case 1 their profile is polynomial and thus the growth of the profile of the subage of height ω 2 is faster than every polynomial.

The reader will note that Case 1 is easier than Case 2. Theorem 6.10 and Lemma 6.11 have a striking consequence for the orbital profile of permutation groups. Theorem 6.12. The orbital profile of an oligomorphic group is either polynomial or faster than every polynomial.

Indeed, if R encodes a permutation group G acting on a set V then the kernel ker(R) of R is the set union of the finite orbits of the one-element subsets of V . If θ G denotes the orbital profile of G then the number of orbits of one-element subsets is at most θ G (1), hence, ker(R) is finite provided that G is oligomorphic.

These results suggest that the link between the height, or the ordinal length, and the growth rate of the profile extends too. This could be the case through hierarchies of functions labelled by ordinals, as is the case with the Wainer-Grzegorczyk hierarchy. But this would require that the arity is bounded. Indeed, if the signature is unbounded the conditions that Age(R) is wqo and ker(R) is finite do not suffice. Our construction of ages yields ages with o(Age(R)) = ω ω and arbitrarily large growth rate ( [START_REF] Pouzet | Ordinal invariant of an age[END_REF]). 6.4. The age algebra of Cameron. The notion of polynomial growth of a profile function is crude. There are at least two others. First, φ R (n) is equivalent to a.n k for some positive real a and nonnegative integer k (in standard notation φ R (n) ≃ a.n k ). Next, φ R (n) is, for n large enough, a quasi-polynomial, that is, a polynomial in n with periodic coefficients. Since φ R is non-decreasing, this will give that φ R (n) ∼ a ⋅ n k . This quasi-polynomiality is better expressed in terms of the generating series of the profile H φ R ∶= ∑ φ R (n). This is a rational fraction of the form (6.1)

P (Z) (1 -Z)(1 -Z 2 )⋯(1 -Z k+1
) where P is a polynomial with integer coefficients (possibly negative).

At the end of the seventies, Cameron, when considering the orbital profile θ G of a permutation group G, conjectured that θ G is asymptotically a polynomial (that is, θ G (n) ≃ a.n k for some a, k when n goes to infinity), whenever it is bounded above by a polynomial. He introduced an algebra, the Cameron algebra, made of the linear combinations of orbits of G. This algebra A(G) is graded and decomposes in a direct sum of subspaces A n made of elements of degree n, the dimension of A n being θ G (n). This implies that the generating series H G of θ G , that is, H G ∶= ∑ θ G (n)Z n , associated to the orbital profile is the Hilbert series of A(G). Macpherson [START_REF] Macpherson | Growth rates in infinite graphs and permutation groups[END_REF], p. 286, conjectured that the algebra A(G) is finitely generated whenever θ G is bounded above by a polynomial. The truth of the conjecture implies that H G is a rational fraction of the form 6.1 above [START_REF] Cameron | Orbits of permutation groups on unordered sets[END_REF], hence θ G is a quasi-polynomial, from which Cameron's conjecture follows. The truth of Macpherson's conjecture was recently proved by Falque and Thiéry [START_REF] Falque | Classification of p-oligomorphic groups; conjectures of Cameron and Macpherson[END_REF], [START_REF] Falque | Macpherson's conjecture holds: the orbit algebra of a permutation group with polynomial profile is finitely generated[END_REF].

Cameron extended his notion of algebra to ages. We reproduce below the presentation given in [START_REF] Pouzet | Some relational structures with polynomial growth and their associated algebras I. Quasi-polynomiality of the profile[END_REF]. Let K be a field of characteristic 0, and V be a set. For n ≥ 0, denote by [V ] n the set of subsets of V of size n, and let K [V ] n be the vector space of maps f ∶ [V ] n → K. The set algebra is the graded connected commutative algebra K

[V ] <ω ∶= ⊕ n K [V ] n , where the product of f ∶ [V ] m → K and g ∶ [V ] n → K is defined as f g ∶ [V ] m+n → K such that: (6.2) (f g)(A) ∶= ∑ (A 1 ,A 2 )∶A=A 1 ⊎A 2 f (A 1 )g(A 2 ) .
Identifying a set S with its characteristic function χ S , elements of the set algebra can be thought as (possibly infinite but of bounded degree) linear combinations of sets; the unit is the empty set, and the product of two sets is their disjoint union, or 0 if their intersection is non trivial.

Let R be a relational structure with base set

V . A map f ∶ [V ] m → K is R-invariant if f (A) = f (A ′
) whenever R ↾A and R ↾A ′ are isomorphic. It is easy to show that the product of two R-invariant maps is again invariant. The K-vector space spanned by the R-invariant maps is therefore a graded connected commutative subalgebra of the set algebra, the age algebra of R, that we denote by KA(R). It can be shown that two relational structures with the same age yield the same algebra (up to an isomorphism of graded algebras); thus the name, coined by Cameron who invented the notion [START_REF] Cameron | The algebra of an age, In Model theory of groups and automorphism groups[END_REF]. If the profile of R takes only finite values, then KA(R) identifies with the set of (finite) linear combinations of elements of Age(R) and, as pointed out by Cameron, φ R (n) is the dimension of the homogeneous component of degree n of KA(R); indeed, define an orbit as the collection of finite subsets A of V such that R ↾A have a given isomorphic type τ , and define an orbit sum as the characteristic function of an orbit; more specifically, the orbit sum of an isomorphism type τ ∈ Age(R) is the characteristic function o τ ∶= ∑ A∈[V ] <ω ∶τ (A)=τ of its representatives in R; then observe that the set of orbit sums form a basis of the age algebra KA(R). By a slight abuse, we sometimes identify τ with its orbit sum to see it as an element of KA(R).

Cameron gave the following illustration of the relevance of the age algebra. Let e ∶= ∑ a∈V {a}, which we can think of as the sum of isomorphic types of the one-element restrictions of R. Let U be the graded linear algebra K[e] = ⊕ ∞ n=0 Ke n . Cameron (see [START_REF] Cameron | The algebra of an age, In Model theory of groups and automorphism groups[END_REF]) proved: Theorem 6.13. If R is infinite then e is not a zero-divisor; namely for any u ∈ KA(R), eu = 0 if and only if u = 0. This result implies that φ R is non-decreasing. Indeed, the image of a basis of the vector space KA(R) n under multiplication by e is a linearly independent subset of KA(R) n+1 .

The relationship between profiles and age algebras is particularly simple for relational structures with bounded profile. These structures (characterized in [START_REF] Fraïssé | Interprétabilité d'une relation par une chaîne[END_REF] for finite signature and in [155, Théorème 1.2] for arbitrary signature, by means of Ramsey's theorem) were defined in Section 5.3. We recall this relationship below. The following theorem links the profile with the age algebra in the context of almost chainable and almost monomorphic relational structures. (e) the age algebra is a finite dimensional linear-module over the linear-algebra K[e], where e ∶= ∑ a∈V {a}; in particular it is finitely generated and Cohen-Macaulay.

As a generalization, Thiéry and I [START_REF] Pouzet | Some relational structures with polynomial growth and their associated algebras I. Quasi-polynomiality of the profile[END_REF] made the following conjecture:

Problem 18. If a relational structure R has finite kernel and profile bounded above by some polynomial then the generating series of the profile is a rational fraction of the form 6.1 above.

The main result of [START_REF] Pouzet | Some relational structures with polynomial growth and their associated algebras I. Quasi-polynomiality of the profile[END_REF] offers a complete solution for relational structures admitting a finite monomorphic decomposition into monomorphic components (see the definition in Subsection 5.6). Theorem 6.15. If R is an infinite relational structure with a finite monomorphic decomposition of dimension k + 1 then the Hilbert series of the profile of the profile is a rational fraction of the form 6.1 above; in particular the profile φ R is eventually a quasi-polynomial of degree k, hence φ R (n) ∼ an k .

Several important structures, for instance, tournaments with polynomial profiles have this property [START_REF] Boudabbous | The morphology of infinite tournaments; applications to the growth of their profile[END_REF], but not graphs. The case of graphs was settled by Balogh et al. [START_REF] Balogh | The unlabelled speed of a hereditary graph property[END_REF] -see Theorem 6.19.

We do not know if Problem 18 has a positive answer for cellular graphs. There are plenty of examples of relational structures whose generating series is a rational fraction of the type described above, but for which the age algebra is not finitely generated ( [START_REF] Pouzet | Some relational structures with polynomial growth and their associated algebras I. Quasi-polynomiality of the profile[END_REF][START_REF] Pouzet | Some relational structures with polynomial growth and their associated algebras II. Finite generation[END_REF]). Problem 19. Is the profile of a relational structure R with finite signature bounded above by some exponential whenever the age of R is well-quasi-ordered under embeddability? Some hypothesis (like the boundedness of the signature) is needed but the answer is not known in the case of graphs. Without this hypothesis, one can build examples of wqo hereditary classes of arbitrarily large profiles (a joint result not yet published [START_REF] Pouzet | Ordinal invariant of an age[END_REF]). The problem above with the hypothesis that Age(R) is hereditarily wqo was proposed in 2003 [START_REF] Pouzet | The order type of the collection of finite series-parallel posets[END_REF].

Paper [START_REF] Pouzet | The profile of relations[END_REF] contains an overview, with a particular emphasis on profiles whose generating function is a rational fraction, the Cameron approach, and some problems linking properties of profiles and of rational languages. A famous result (Chomsky-Schützenberger's theorem, see [START_REF] Berstel | Les séries rationnelles et leurs langages[END_REF]) asserts that the generating series of a regular language is a rational function (see [START_REF] Abdi | An example of Tateno disproving conjectures of Bonato-Tardif, Thomassé and Tyomkyn[END_REF]). This result is not far away from our considerations. Indeed, if A is a finite alphabet, with say k letters, and A * the set of words over A, then each word can be viewed as a finite chain coloured by k colors. Hence A * can be viewed as the age of the relational structure R made of the chain Q of rational numbers divided into k colors in such a way that, between two distinct rational numbers, all colors appear. As pointed out by Cameron [START_REF] Atminas | Labelled induced subgraphs and well-quasi-ordering[END_REF], the age algebra Q. Age(R) is isomorphic to the shuffle algebra 6.5. The case of graphs and other binary structures. During the last fifteen years, many results concerning the enumeration of classes of binary relations have appeared. Among them, it is worth noting the results of Albert and Atkinson [START_REF] Albert | Simple permutations and pattern restricted permutations[END_REF], Balogh et al. [START_REF] Balogh | The unlabelled speed of a hereditary graph property[END_REF][START_REF] Balogh | Hereditary properties of partitions, ordered graphs and ordered hypergraphs[END_REF][START_REF] Balogh | Hereditary properties of ordered graphs[END_REF][START_REF] Balogh | Hereditary properties of tournaments[END_REF] (see [START_REF] Klazar | Overview of general results in combinatorial enumeration, in Permutation patterns[END_REF][START_REF] Vatter | Permutation classes[END_REF] for an overview). These results as well as the recent result by Albert et al. [START_REF] Albert | Rationality for subclasses of 321-avoiding permutations[END_REF] proving that the generating series of the profile of a hereditary class of bichains is a rational fraction supports a positive answer to Problem 19 above.

The class of graphs provides interesting examples of profiles.

Examples 6.16. In each of the following G is a graph.

(1) φ G (n) = 1 for every n ≤ |V (G)| if and only if φ G (2) ≤ 1, that is, G is a clique or an independent set. (This is trivial.)

(2) φ G is bounded if and only if G is "almost constant" in the Fraïssé's terminology, that is, there is a finite subset F G of vertices such that two pairs {x, y} and {x ′ , y ′ } of vertices having the same intersection on F G are both edges or both non-edges.

(3) If G is the direct sum of infinitely many edges, or the direct sum K ω ⊕ K ω of two infinite cliques, then φ G (n) = ⌊ n 2 ⌋ + 1 and H φ G = 1

(1-x)(1-x 2 ) . (4) Let G be the direct sum K (1,ω) ⊕ K ω of an infinite wheel and an infinite independent set, or the direct sum K ω ⊕ K ω of an infinite clique and an infinite independent set. Then φ G (n) = n. Hence,

H φ G = 1 + x (1-x) 2 = 1+x 3
(1-x)(1-x 2 ) . ( 5) Let G be the direct sum of infinitely many k-element cliques or the direct sum of k infinite cliques.

Then φ G (n) = p k (n) ≃ n k-1 (k-1)!k! and H φ G = 1

(1-x)⋯(1-x k ) . ( 6) If G is either the direct sum of infinitely many infinite cliques, or an infinite path, then φ G (n) = p(n), the partition function.

(7) Let C ∶= (E, ≤) be a chain and K C, 1 2 be the graph whose vertex set is 2 × E and whose edge set is {{(0, i), (1, j)} ∶ i < j in C}. Such a graph is an half-complete bipartite graph. If C is infinite, then 2 n-2 ≤ φ K C, 1 2 (n) ≤ 2 n-1 [START_REF] Macpherson | Growth rates in infinite graphs and permutation groups[END_REF], hence its growth is exponential. In fact, one can check that: H K C, 1 2 = 1-2x-x 2 +3x 3 -x 4

(1-x)(1-2x)(1-2x 2 ) = 1+x+2x 2 +3x 3 +6x 4 +10x 5 +20x 6 +36x 7 +72x 8 +136x 9 +O(x 10 ). Cellular graphs introduced in Subsection 5.4.2 enter in the picture.

Theorem 6.17.

[157] (Theorem 2.14 p. 246) The profile of a graph is bounded by a polynomial if and only if this graph is cellular.

A straightforward computation shows that the profile of a cellular graph is bounded by a polynomial. The converse follows directly from Theorem 6.6 ensuring that a graph whose profile is bounded by a polynomial must be almost multichainable and from Lemma 6.18 below (see Lemma 2.15 p.246 in [START_REF] Pouzet | The profile of relations[END_REF] for a proof). Lemma 6.18. The growth of the profile of an almost multichainable graph which is not cellular is at least exponential.

A more precise result was obtained by Balogh, Bollobás, Saks and Sós [START_REF] Balogh | The unlabelled speed of a hereditary graph property[END_REF]. Theorem 6.19. If G is an undirected graph then either φ G is a quasi-polynomial of degree k, in which case φ G (n) ≃ a.n k , or φ G (n) is bounded below by p(n), the partition function which counts the number of partitions of the integer n.

This improves Macpherson's result [START_REF] Macpherson | Growth rates in infinite graphs and permutation groups[END_REF] asserting that in the latter case of the result above φ G is asymptotically bounded below by any function of the form exp(n 1 2 -ε ) with ε > 0. For ordered graphs the situation is a bit different. Balogh, Bollobás and Morris [START_REF] Balogh | Hereditary properties of ordered graphs[END_REF] proved that if C is a hereditary class of finite ordered graphs then its profile φ C is either polynomial or is ranked by the Fibonacci functions (see Klazar [START_REF] Klazar | Overview of general results in combinatorial enumeration, in Permutation patterns[END_REF] for recent developments). Using the notion of monomorphic decomposition, Oudrar [START_REF] Oudrar | Hereditary classes of ordered binary structures[END_REF] proves that the class S of ordered binary structures which do not have a finite monomorphic decomposition has a finite basis (a subset B such that every member of S embeds some member of B). In the case of ordered reflexive directed graphs, the basis has 1242 members and the profile of their ages grows at least as fast as the Fibonacci function. From this, Oudrar gets a dichotomy result, recently extended to ordered relational structures [START_REF] Oudrar | Ordered structures with no finite monomorphic decomposition[END_REF]. We conclude this section by mentioning that a new parameter, twin width, has been introduced and linked to the growth rate of the profile of ordered graphs; see, for instance, [START_REF] Bonnet | Twin-width IV:ordered graphs and matrices[END_REF][START_REF] Geniet | First order logic and twin-width in tournaments and dense oriented graphs[END_REF].

THE NUMBER OF WQO HEREDITARY CLASSES OF FINITE STRUCTURES

Due to Corollary 3.11, the number of wqo hereditary classes of finite structures is the number of wqo ages. As mentioned in Corollary 2.24, if the maximum of the signature is 1 and the number of unary symbols is finite, then Ω µ is wqo. Hence all ages included in Ω µ are wqo and there are only countably many. The situation is radically different if the maximum of the arity is at least 2. First, Ω µ is not wqo and next, the number of ages has the cardinality of the continuum (see Lemmas 2.21 and 2.23). I show below that there are 2 ℵ 0 wqo ages. This result (in a slightly different form) is in [START_REF] Pouzet | Sur la théorie des relations[END_REF] with a proof in [START_REF] Pouzet | Graphs containing finite induced paths of unbounded length[END_REF]. It is far from being immediate. It goes through a coding of binary structures by a particular kind of infinite words on {0, 1}, the so-called uniformly recurrent words. These binary structures and their encoding are simple to define, and are given below. After that, we will define another encoding leading to graphs.

To each infinite word µ ∶= (u n ) n∈N on 0 and 1 let us associate the binary structure R µ ∶= (N, ρ µ ) where ρ µ∶ = {(n, n + 1) ∶ n ∈ N} ∪ {(n, n) ∶ µ(n) = 1}. Hence, R u is a directed path with some loops. Let us also associate the set Fac(µ) of finite factors of µ and order it via the factor ordering. It is immediate to see that if µ and µ ′ are two infinite words then (7.1) Fac(µ) ⊆ Fac(µ ′ ) if and only if Age(R µ ) ⊆ Age(R µ ′ ).

The words we choose are the uniformly recurrent words which appear in symbolic dynamics. Let us give the definition first: an infinite word µ on {0, 1} is uniformly recurrent if for every nonnegative integer n there is some nonnegative integer m such that each factor of µ of length n is a factor of every factor of µ of length m. There are plenty of examples. For instance, consider Sturmian words: infinite words u such that for each nonnegative integer, the number Fac n (µ) of factors of µ with length n is n + 1. These words can be obtained as billiard words or rotation words. Two Sturmian words may have the same set of factors, but there are 2 ℵ 0 Sturmian words, no two of which have the same set of finite factors. As an example, take the Sturmian words with different slopes α ∈]0, 1[, where the slope is the limit as n goes to infinity of the proportion of 1's in factors of length n (see Fogg [START_REF] Fogg | Substitutions in dynamics, arithmetics and combinatorics[END_REF], Chapter 6).

Since there are 2 ℵ 0 uniformly recurrent words with distinct sets of finite factors, Theorem 7.1 follows from the following lemma. Lemma 7.2. For every uniformly recurrent word µ, Age(R µ ) is wqo.

The proof is based on two facts: (a) Fac(µ) is a Jónsson poset, hence is wqo, whenever µ is uniformly recurrent; (b) Age(R µ ) is the image by an order preserving map of the set Fac(µ) * of finite words over Fac(µ) viewed as an alphabet, hence, by Higman's theorem, is wqo. Here are some details.

Let us say that a subset J of the set of finite words on {0, 1} is inexhaustible if it is not reduced to the empty word and if for every v ∈ J there is some word w such that vwv ∈ J. A word µ is recurrent if is is not the empty word and every finite factor occurs infinitely often; this amounts to the fact that Fac(µ) is inexhaustible. We will use the following result, cf. Theorem 5 of [START_REF] Pouzet | Graphs containing finite induced paths of unbounded length[END_REF].

Theorem 7.3. Let µ be an infinite word on {0, 1} with domain N. Then, the following properties are equivalent:

(i) µ is uniformly recurrent;

(ii) µ is recurrent and Fac(µ) is wqo;

(iii) Fac(µ) equipped with the factor ordering is a countable Jónsson poset.

The equivalence between (i) and (iii) follows from the equivalence between (i) and (iii) of Theorem 5.2. Implication (ii) ⇒ (iii) is detailed in [START_REF] Pouzet | Graphs containing finite induced paths of unbounded length[END_REF].

Fact (a) is contained in Theorem 7.3. For the proof of (b), let µ be a uniformly recurrent word with domain N and F be a finite subset of N. This set is a finite union of disjoint intervals of N, say F 0 , . . . , F m , two consecutive intervals F i , F i+1 being separated by at least some a i ∈ N. To each F i , one may associate the word v i corresponding to the restriction of u to the interval F i and to F the finite sequence S F of finite words v 0 , . . . , v m , each v i being the restriction of µ to F i . Since the collection of finite factors of µ is wqo for the factor ordering, Higman's theorem on words (Theorem 3.18) says that the collection of words over the alphabet Fac(µ), thus the collection of words of the form S F , is wqo. Since, as it is easy to check, S F ≤ S F ′ implies that R µ ↾ F embeds in R µ ↾ F ′ , one gets that Age R µ is wqo.

One may refine Theorem 7.1 as follows.

Proposition 7.4. There is a hereditary class of finite relational structures C such that Id(C) has the cardinality of the continuum and every member of Id(C) is wqo (and in fact bqo).

Proof. Let µ be a uniformly recurrent word which is non periodic and let F be the set of words µ ′ which have the same set of finite factors as µ. Since µ is uniformly recurrent and is not periodic, F has the cardinality of the continuum. Furthermore, each Age(C s ) is bqo.

The graphs constructed above are directed with loops. One may replace them by undirected graphs, with loops, see the proof in [START_REF] Pouzet | Graphs containing finite induced paths of unbounded length[END_REF]. But to get rid of loops, another construction is needed. Lemma 7.15. If µ is finite then P rim µ is level-finite.

Proof. Suppose for a contradiction that there exists an integer n ≥ 0 such that the level P rim µ (n) of P rim µ is infinite and choose n smallest with this property. Define C ∶= {R ∈ Ω µ ∶ R < S for some S ∈ P rim µ (n)}.

Then C is a hereditary class of Ω µ containing only finitely many prime structures. According to (2) of Theorem 3.23, C has only finitely many bounds. This is not possible since the elements of P rim µ (n) are bounds of C.

Proof of Theorem 7.14. Let C be a hereditary class of Ω µ such that J ∶= P rim µ (C) is infinite. Since P rim µ is level-finite, Lemma 5.3 ensures that J contains an initial segment D which is Jónsson. According to Theorem 7.13, ↓ D is minimal prime. This completes the proof. ◻

The following result gathers the main properties we know about minimal prime structures.

Theorem 7.16. Every minimal prime hereditary class C is the age of some prime structure. Furthermore, if the signature is finite, this age is wqo. If it is exhaustible then it is hereditarily wqo and hence almost inexhaustible.

The first and second part form Theorem 19 of [START_REF] Oudrar | Minimal prime ages, words and permutation graphs[END_REF]. The third part is Lemma 23 and Theorem 20 of [START_REF] Oudrar | Minimal prime ages, words and permutation graphs[END_REF].

We give a proof.

Proof of the first and second part of Theorem 7.16. Let C be a minimal prime hereditary class. We first prove that it is the age of a prime structure. It follows from Theorem 7.13 that C =↓ D where D is Jónsson. Since D is Jónsson, it is up-directed. Thus C is an age. Since D is up-directed and countable, it contains a cofinal sequence R 0 ≤ R 1 ≤ . . . < R n ≤ . . .. We may define the limit R of these R n . Since the R n 's are prime, R is prime and Age(R) = C. Next we prove that C is wqo. Since D is Jónsson, it is wqo. To prove that C is wqo, let R ∈ C, consider C ∖ (↑ {R}), and observe that it is enough to prove that C ∖ (↑ {R}) is wqo by embeddability. Indeed, an antichain that contains R must be in C ∖ (↑ {R}). Now to prove that C ∖ (↑ {R}) is wqo, we note that since C ∖ (↑ {R}) is a proper hereditary class in C, it contains only finitely many primes. It follows from Theorem 3.23 that C ∖ (↑ {R}) is wqo. ◻

For the proof of the third part, let C be a class of finite binary structures S ∶= (F, (ρ i ) i∈I ) with F finite and a given finite signature µ. Denote by C +1 the class of S ∶= (F, (ρ i ) i∈I ) such that there is some a ∈ F such that S ↾F ∖{a} ∈ C.

The following lemma is Proposition 5.32 p. 105 of [START_REF] Oudrar | Sur l'énumération de structures discrètes: une approche par la théorie des relations[END_REF] and Theorem 4.5 page 20 of [START_REF] Brignall | Labelled well-quasi-order for permutation classes[END_REF]. A similar fact, but not explicit, appears in the proof of Theorem 4.24, p. 267 of [START_REF] Pouzet | The profile of relations[END_REF]. For the reader's convenience, we give a proof. Lemma 7.17. Let C be a hereditary class of binary structures with a finite signature. If the members of C are not necessarily finite and if these members when labelled by any better-quasi-order form a well-quasi-order, then C +1 has the same property. If C is made of finite structures and is hereditarily well-quasi-ordered, then C +1 is hereditarily well-quasi-ordered.

Proof. Let I be such that each S ∈ C is of the form S ∶= (F, (ρ i ) i∈I ). Let W be a wqo. By hypothesis, the set (2 × 2 × 2) I is finite, hence, it is wqo by equality. The direct product W ′ = W × (2 × 2 × 2) I is wqo. We encode the members of C +1 labelled by W , by members of C labelled by W ′ . Indeed, for each S ∶= (F, (ρ i ) i∈I ) ∈ C +1 we select a ∈ F such that S ↾F ∖{a} ∈ C and we label S ↾F ∖{a} by the map g a defined for x ∈ F ∖ {a} by g a (x) ∶= (ρ i (a, x), ρ i (x, a), ρ i (a, a)) i∈I . Now if f is a labelling of F in W , we associate the labelling f ′ of F ∖ {a} by setting f ′ ∶= (f ↾F ∖{a} , g a ). By construction, if S, S ′ ∈ C, an embedding h from the labelled structure S ↾F ∖{a} in the labelled structure S ′ ↾F ′ ∖{a ′ } will extend to an embedding of the labelled structure S in the labelled structure S ′ with a mapped to a ′ . The conclusion follows. We have seen with Theorem 7.23 and Proposition 7.24 that the ages in L are minimal prime. The converse is based on the theorem of Chudnovsky et al. [START_REF] Chudnovsky | Unavoidable induced subgraphs in large graphs with no homogeneous sets[END_REF] noted at the beginning of this subsection. One can also use an infinite version due to Malliaris and Terry [START_REF] Malliaris | On unavoidable-induced subgraphs in large prime graphs[END_REF].

In [START_REF] Chudnovsky | Unavoidable induced subgraphs in large graphs with no homogeneous sets[END_REF], Theorem 1.2, the authors prove that for every integer n ≥ 3 there is an integer N such that every prime graph with at least N vertices contains one of the following graphs or their complements as an induced subgraph:

(1) the graph obtained from K 1,n by subdividing every edge once;

(2) the line graph of K 2,n ;

(3) the line graph of the graph in (1);

(4) the half-graph of height n;

(5) a prime graph induced by a chain of length n; and, (6) two particular graphs obtained from the half-graph of height n by making one side a clique and adding one vertex. With this result, the proof that there are not other minimal prime ages of graphs than those of L is a simple exercice. Indeed, let C be a minimal prime age of graphs. Since C contains infinitely many prime graphs, it contains an infinite set D of prime graphs belonging to one of the six types described in the theorem. Since C is minimal prime, ↓ D = C. If D contains infinitely many primes in (1), respectively, in (2), in (3), in (4), then, via an inspection of the ages in M (and in L), D ⊆ Age(G 3 ), respectively, D ⊆ Age(G comp 0 ), D ⊆ Age(G 4 ), D ⊆ Age(G 1 ). And, if D contains infinitely many primes in [START_REF] Atminas | Labelled induced subgraphs and well-quasi-ordering[END_REF], then either D ⊆ Age(G 5 ) or D ⊆ Age(G 6 ). Since the ages of those G i are minimal prime then the previous inclusions yield the equality C = Age(G i ). The last possibility for D is that D is made of chains, that is, graphs in [START_REF] Assous | Jónsson posets[END_REF]. In that case C is the age of some G µ for µ uniformly recurrent.

EXTENSIONS OF LAVER'S THEOREM

Sierpinski [START_REF] Sierpinski | Sur les types d'ordre des ensembles linéaires[END_REF] proved the existence of an infinite strictly decreasing sequence of subchains of the chain of real numbers. Fraïssé [START_REF] Fraïssé | Sur la comparaison des types d'ordres[END_REF] conjectured that there is no infinite strictly decreasing sequence of countable chains and he made several other conjectures about the embeddability between countable chains. The fact that the class of countable chains contains no infinite strictly decreasing sequence and no infinite antichain, that is, the class of countable chains is wqo, was known under the name of Fraïssé's conjecture. This conjecture was positively solved by Laver [START_REF] Laver | On Fraïssé's order type conjecture[END_REF] in 1971, using the Hausdorff decomposition theorem of scattered chains and the theory of better-quasi-ordering, due to Nash-Williams. Laver proved more strongly that the class of chains which are countable unions of scattered chains is bqo, hence wqo under embeddability. For a detailed exposition of Fraïssé's conjectures, bqo and Laver's proof, see Rosenstein's book [START_REF] Rosenstein | Linear orderings[END_REF], Chapters 9 and 10.

A consequence of the well-foundedness of the class of countable chains is the fact that every countable chain is a finite lexicographic sum of additively indecomposable chains. (Recall that a chain α is additively indecomposable if for every decomposition α = α 1 + α 2 either α ≤ α 1 or α ≤ α 2 .) The proof is a typical argument about well-founded sets. Indeed, if the property fails for some chain, there is some minimal chain α for which it fails. Decompose this chain into two parts α 1 and α 2 such that α = α 1 + α 2 . If α 1 < α and α 2 < α then, by the minimality of α, α 1 and α 2 are finite sums of additively indecomposable chains and hence α is a finite sum, contradicting the choice of α. Hence, either α ≤ α 1 or α ≤ α 2 and thus α is additively indecomposable, again contradicting the choice of α. Thus no such α can exist.

Laver also showed that every additively indecomposable scattered chain can be obtained from the chains 0 and 1 by means of regular unbounded sums. (A lexicographical sum ∑ µ<κ α µ or ∑ µ<κ * α µ of scattered types is regular and unbounded if κ is an infinite regular ordinal and for every µ < κ, the set {ν ∶ α µ ≤ α ν } has cardinality κ). In [START_REF] Laver | An order type decomposition theorem[END_REF], he gives a description of indivisible scattered chains, called strongly indecomposable in his paper. (Recall that a chain α is indivisible if for each partition of its domain into two parts A and B one of the parts embeds α). The bqo character of C is not enough. Examples of bqo ages A for which the collection of countable structures with this given age is not wqo were given in [START_REF] Pouzet | Sur la théorie des relations[END_REF]. In fact, there are 2 ℵ 0 bqo ages A of graphs such that the class A ω of graphs with age included in A is not well-founded. The ages of graphs G α associated to non-periodic and uniformly recurrent words α in Subsection 7.2 are such structures [START_REF] Oudrar | Minimal prime ages, words and permutation graphs[END_REF], Remark 6, p. 23.

Here, we give some positive examples. A poset is series-parallel (see Subsection 3.3.1) if it does not embed an "N ", that is, the 4-element poset {a, b, a ′ , b ′ } with exactly the strict comparabilities a < a ′ , b < b ′ and b < a ′ .

Thomassé [START_REF] Thomassé | On better-quasi-ordering countable series-parallel orders[END_REF] proved:

Theorem 8.1. The class of countable series-parallel posets is wqo (and in fact bqo).

An extension to binary structures was obtained independently by Delhommé [START_REF] Delhommé | Nicely BQO grounded categories and 2-structures[END_REF] and McKay [START_REF] Mckay | Better-Quasi-Orders: Extensions and Abstractions[END_REF]:

Theorem 8.2. If a hereditary class C of finite binary structures of finite arity contains only finitely many indecomposable members then the class of countable structures whose age is included into C is bqo.

As for Thomassé's result, the proof relies on a result of Corominas on labelled trees [START_REF] Corominas | On better-quasi-ordering countable trees[END_REF] and the extension of Gallai decomposition theory [START_REF] Gallai | Transitiv orientbare graphen[END_REF] to 2-structures [START_REF] Ehrenfeucht | The theory of 2-structures. A framework for decomposition and transformation of graphs[END_REF], notably the properties of tree-decompositions (see [START_REF] Harju | Decomposition of infinite labeled 2-structures[END_REF], [START_REF] Courcelle | The modular decomposition of countable graphs. Definition and construction in monadic secondorder logic[END_REF]).

Problems 22.

(1) If the set Ind(C) of indecomposable structures contained in a hereditary class C of finite binary structures is hereditarily bqo then the class of countable structures whose age is a subclass of C is bqo.

(2) If a hereditary class C of finite structures, not necessarily binary, is hereditarily bqo then the class of countable structures whose age is a subclass of C is bqo (and even hereditarily bqo).

CONCLUDING REMARKS

Lemma 2 . 5 .

 25 Each hereditary class C included in Ω µ is determined by its bounds. More precisely, C = F orb(B) where B ∶= Bound(C).

Theorem 3 . 7 .

 37 If P is wqo then o(P ) = h(P, I(P )). Example 3.8. If A is an alphabet made of p letters then the set A * of words over A, ordered by the subword ordering, is wqo and o

Proposition 3 . 10 .

 310 Let C be a hereditary class of finite structures. Then the following properties are equivalent: (a) C contains no infinite antichain; (b) C contains at most countably many hereditary subclasses; (c) the set I(C) of hereditary subclasses of C ordered by set inclusion is well-founded; (d) the powerset ℘(N) ordered by set inclusion does not embed in the collection I(C) of hereditary subclasses of C ordered by inclusion. Proof. The proof is straightforward: (a) and (b) are equivalent. Indeed, if (a) holds then C is level-finite, hence C is at most countable. According to (ii) of Theorem 3.1, each final segment F of C is finitely generated, hence the set F(C) of these final segments is at most countable. Since I(C) is dually isomorphic to F(C), it is at most countable and (b) holds. Conversely, if (b) holds, then no infinite antichain can exist, otherwise the collection of final segments of C generated by subsets of such an antichain will be uncountable and thus F(C) will be uncountable and I(C) too. That (a) is equivalent to (c) follows from the equivalence of (i) and (iii) of Theorem 3.1.

Proof.

  For the second part, let D ∶= Ind(C) be the set of prime members of C. Then C is included into ∑ D, the least subclass of Ω µ containing D and stable by lexicographical sum indexed by members of D. Since D is finite then, according to the first part of Theorem 3.23, ∑ D is wqo. Hence, C has finitely many bounds in ∑ D. If k is the maximum of the cardinality of members of D the bounds of ∑ D have size at most k + 2.

3. 5 .

 5 From hereditary classes to ideals. From Corollary 3.11, a part of the study of wqo hereditary classes of finite structures can be reduced to the study of wqo ages. Problem 4. Are the previous equivalences in Proposition 3.10 still true if we replace C by an age and the hereditary subclasses of C by ages (see p. 167,[START_REF] Pouzet | Condition de chaîne en théorie des relations[END_REF])?

Theorem 3 .

 3 35. g(S, A) is finite, in fact |g(S, A)| ≤ 2 | ker(S,R)| where | ker(S, R)| is the number of elements of ker(S, R) for any relational structure R having age A.Let A ′ ⊆ A be two ages and [A ′ , A] be the set of ages B such that A ′ ⊆ B ⊆ A. Theorem 3.35 is the key of the following result (see Proposition V-3.2 of[START_REF] Pouzet | Condition de chaîne en théorie des relations[END_REF], Theorem 3.11 and Corollary 3.14 of[START_REF] Pouzet | Sandwiches of ages[END_REF]).

Theorem 3 . 36 .

 336 Let A ′ ⊆ A be two ages. Then the following properties are equivalent (i) A ′ and some S ∈ A generate A.(ii) The interval [A ′ , A] ordered by set inclusion has finite height. (iii) The interval [A ′ , A] is finite.

Corollary 3 . 38 .

 338 Let A be an age such that Id(A) is well-founded; (a) If A is inexhaustible, then H(A) is a limit ordinal; (b) If A is a strictly increasing union of inexhaustible ages then H(A) ≥ ω 2 .

  (α) (C)). If A is minimal, then Id(A) is well-founded and H(A) ≤ ω.α, where H(A) ∶= h(A, Id(C)) as defined by formula (1.1); furthermore, if A has finitely many bounds in C then it has Cantor-Bendixson rank α.(2) If A has Cantor-Bendixson rank α, that is A / ∈ Id (α+1) (C), then Id(A) is well-founded and H(A) < ω ⋅ (α + 1).

Problem 7 .

 7 Is ω.rank(A, Id(A)) ≤ H(A) whenever the signature of members of A is finite? If Id(C) is topologically scattered, then every A ∈ Id(C) has a Cantor-Bendixson rank. Hence (2) of Proposition 3.39 yields: Theorem 3.40. If the set Id(C) of ages of a level-finite hereditary class C of finite structures is topologically scattered for the product topology then it is well-founded with respect to set inclusion.

( 1 )

 1 An age A ∈ C has a Cantor-Bendixson rank n in Id(C) if and only if Id(A) is well-founded and ω ⋅ n ≤ H(A) < ω ⋅ (n + 1). (2) If Id (n) (C) is nonempty then the set of isolated points it contains is dense. Proof. Let A ∈ C with Cantor-Bendixson rank n in Id(C). According to (2) of Proposition 3.39, Id(A) is well-founded and H(A) < ω ⋅ (n + 1). According to Theorems 5.34 and 5.42 given in Subsection 5.4,

Lemma 4 . 7 .

 47 If a hereditary class C of (undirected loopless) graphs is 1-wqo, it has finitely many bounds.The proof is easy. In each boundB of C, pick a vertex x B . Set D B ∶= B ↾ V (B) ∖ {x B } and D B ∶= D B ⋅ X B where X B ∶= {y ∈ V (B) ∶ y / = x B and {x B , y} ∈ E(B)}. Clearly, if B ′ is another bound and f is an embedding of D B into D B ′ then the map f extending f by setting f (x b ) ∶= x B ′ is an embedding of B into B ′ .Hence, if B is an infinite set of bounds of C, the set {D B ∶ B ∈ B} is an infinite antichain of C[START_REF] Abdi | An example of Tateno disproving conjectures of Bonato-Tardif, Thomassé and Tyomkyn[END_REF] , which then is not wqo.

Problem 11 .

 11 Find a common root to Lemmata 4.7 and 4.8. The proof of Lemma 4.8 can be extended to non-binary relational structures. Theorem 4.9. Let C be a hereditary class of relational structures with arity bounded by m. If the class C

5. 1 . 1 .

 11 Jónsson posets. Definition 5.1. A poset P is a Jónsson poset if it is infinite and every proper initial segment has a strictly smaller cardinality than P .

Proposition 5 . 19 .

 519 Let µ be a finite signature. (a) Each infinite hereditary subclass C of Ω µ contains the age of an infinite chainable structure. (b) The number of such ages is finite. (c) For each T ∈ Ω µ there is a minimal infinite age containing T . (d) Each minimal infinite age containing T ∈ Ω µ is the age of an almost chainable structure.

5. 4 .

 4 Almost multichainability. Finite ages and ages of infinite almost chainable structures are the first layers of a decomposition of the set of ages. The ages we encounter next are those of almost multichainable relational structures.

  is val-finite by Theorem 5.47. Apply Lemma 5.46 to get two finite subsets F and G of V ∖ A and A, respectively, such that the local isomorphisms of R ↾A and R ↾V ∖A that extend by the identity on F and on G are those that extend the identity on V ∖ A and A, respectively.SetM ∶= R ⋅ a 1 ⋅ ⋯ ⋅ a k ⋅ b 1 ⋅ ⋯ ⋅ b ℓwhere a 1 , . . . , a k and b 1 , . . . , b ℓ are constants enumerating F and G, respectively.

Theorem 6 . 14 .

 614 Let R be a relational structure on an infinite domain V . Then, the following properties are equivalent: (a) the profile of R is bounded; (b) R is almost monomorphic; (c) R is almost chainable; (d) the Hilbert series is of the following form, with P (Z) ∈ N[Z]: H R = P (Z) 1 -Z ;

( 8 ) 1 2 1 2 2 (

 8112 Let KC, be the graph obtained from K C, by adding all possible edges between vertices of the form (1, i), for i ∈ E. Then φ KC, 1 n) = 2 n-1 .

7. 1 .

 1 The number of wqo ages of binary relations. Theorem 7.1. There are 2 ℵ 0 wqo ages of binary relations.

  For each s ∈ F , let C s ∶= (N, ρ µ , 0) be the structure made of the binary relation ρ µ on N defined by ρ µ (n, m) = 1 if either m = n+1 or m = n and µ(n) = 1, and 0 is the unary relation so that 0(n) = 1 if and only if n = 0. Let C ∶= ⋃ s∈F Age(C s ). Then Id(C) = ⋃ s∈F Id(Age(C s )).

FIGURE 4 .

 4 FIGURE 4.

FIGURE 5 .

 5 FIGURE 5.

  Fraïssé, [74], p. 249, asked: Problem 21. Which hereditary classes C of finite structures are such that the class C ω (made of countable structures R such that Age(R) ⊆ C) is wqo?

  2.3. Hereditary classes. A class C of structures is hereditary if it contains every structure that can be embedded into some member of C. Hence, hereditary classes are initial segments of the class of relational structures quasi-ordered by embeddability. Let us start with examples of hereditary classes of finite structures. Bounds. A bound of a hereditary class C of finite structures is any element of Ω µ ∖ C that is minimal w.r.t. embeddability. Hence, R is a bound of

	Examples 2.4. The following classes of finite structures are hereditary:
	(1) chains;
	(2) posets;
	(3) ordered forests (two elements which have a upper bound are comparable);
	(4) posets coverable by at most n chains, where n is a fixed nonnegative integer;
	(5) posets of Dushnik-Miller dimension at most n;
	(6) posets of jump-number at most n (the order can be made linear by adding at most n comparabilities);
	(7) graphs;
	(8) comparability graphs;
	(9) planar graphs;
	(10) n-colourable graphs;
	(11) perfect graphs (graphs such that the chromatic number of every induced subgraph is the size of the
	largest clique of that subgraph).
	The most important problem about a class C of finite structures is to decide whether a finite relational
	structure R belongs to C or not. When C is hereditary, this question leads to the notion of bound.
	2.4.

  This is a purely order-theoretic fact. To prove it, suppose that C has no greatest element, and consider an enumeration of the elements of C in a sequence c 0 , . . . , c n , . . . . Define t 0 , . . . , t n , . . . inductively by setting t 0 ∶= c 0 and t n being defined, t n+1 be an upper bound of t n and c n+1 . Now, with this fact at hand, if C has a greatest element t, then it is the age of any relational structure of type t. Otherwise, construct a sequence R 0 , R 1 , . . . , R n , . . . of relational structures such that, for each nonnegative integer n, R n has type t n and R n is a restriction of R n+1 . Let R ∞ ∶= ⋃ n≥0 R n . Then, clearly C is the age of R ∞ . It suffices to show that C contains a countable ideal and then to apply Theorem 2.7. In fact, if C is an ideal, the proof of Theorem 2.7 yields a strictly increasing sequence t 0 < t 1 < ⋯ < t n ⋯ of elements of C. It generates an ideal which is countable since ↓ t n is finite for each n. If C is infinite and level-finite it has infinitely many levels. König's Theorem asserts that C contains an infinite chain going through all the levels. This chain generates an ideal. Since each principal initial segment is finite, this ideal is countable.

	Theorem 2.8. Let C be an infinite ideal of finite structures or an infinite hereditary class of finite structures
	which is level-finite. Then C contains a countable age.
	Proof. Proposition 2.9. If the signature µ is finite, Ω µ is level-finite and hence countable. Thus every ideal of Ω µ
	is the age of a countable structure. Moreover, no matter whether µ is finite or not, Ω µ is an ideal and in fact
	an age.

  Hence we have the following dichotomy result: Lemma 3.29. For every level-finite age C, either C does not embed [ω] <ω or C is equimorphic to [ω] <ω . 3.5.1. A central question about ages. It is a tempting and old conjecture that all statements in Theorem 3.28 are equivalent when C is an age. Conjecture 1. An age C does not embed [ω] <ω if and only if it contains no infinite antichain, that is, C is wqo.

  Theorem 5.8. Let C be a hereditary class of finite structures. Then the following properties are equivalent.(i) C is an infinite ideal of Ω µ containing no proper infinite ideal;(ii) C is Jónsson, that is, o(C) = ω;(iii) C is the age of a countable chainable structure.

Proof. (iii) ⇒ (ii). If C is the age of a chainable relational structure then it is totally ordered by embeddability and has order type ω, thus it is Jónsson. (ii) ⇒ (i). If C is Jónsson it does no contain any proper infinite initial segment hence it does not contain any proper infinite ideal. (i) ⇒ (iii). According to Theorem 2.8, C is the age of a countable structure, say R ∶= (V, (ρ i ) i∈I ). If C contains no proper infinite subage, then Age

  interprets freely R ′ ↾V ′ . It follows that R ↾V ′ is almost multichainable with the same age as R. ◻ An immediate application of Theorem 4.10 is this: Theorem 5.39. The age of an almost multichainable relational structure with a finite signature has finitely many bounds. 5.5. Ages of height strictly smaller than ω 2 . The central result of this subsection is Theorem 5.42 asserting that ages of height strictly smaller that ω 2 are ages of almost multichainable structures. A special case is given by the following consequence of Proposition 5.26 (see Corollary 4.22 of [157]). Proposition 5.40. If the age A of a denumerable structure R is inexhaustible and level-finite, then A is the union of an increasing sequence (A n ) n∈N where each A n is the age of a product of some finite relational structure by an infinite chain. If furthermore A has an height and H(A) < ω 2 then A is the age of a product of a finite relational structure by a chain.

  5.40 extends as follows:Proposition 5.41. If an age A is almost inexhaustible and level-finite, then this is the age of the union S of an increasing sequence (S n ) n∈ N of almost multichainable structures each having the same kernel as S. Consequently, if A is not the age of an almost multichainable structure, it contains ages of height ω ⋅ n for each integer n. Furthermore, if A has a height and H(A) < ω 2 , then A is the age of an almost multichainable structure.Proof. If R is any structure with age A, the age of a reduction of R over its kernel, say R ker(R) , is inexhaustible. Then there is some R ′ with the same age as R such that R ′ ker(R ′ ) is a product of some structure R ′′ by an infinite chain. Apply Proposition 5.40 to R ′ ker(R ′ ) . Let A be the age of a relational structure with finite signature. If H(A) < ω 2 then A is the age of an almost multichainable relational structure.

	Theorem 5.42.

  5.5.3. Ages of multichainable relational structures of height at least ω 2 . Theorem 5.51. If an age of a relational structure with finite signature has no height then it contains an age of height ω 2 . Proof. If an age A has no height, A is not wqo. According to Lemma 3.12, A contains a wqo age B with infinitely many bounds in A. Being wqo, this age has a height. According to Theorems 5.34 and 5.42, its height cannot be less than ω 2 . Corollary 5.52. Every age of height at most ω 2 is wqo.Theorem 5.53. Let n and p be two integers. There are finitely many ages of height ω n + p in a given finite signature.Proof. One shows by induction on ω.n + p (and some effort) that the parameters of almost multichainable relational structures of height ω n +p are bounded, hence by Lemma 2.18 there are only finitely many. This result is the best possible. If the arity is at least 2, there are infinitely many ages of height ω 2 and in fact a continuum (see Subsection 7.2).Let A be an age. Let Id(Almulti(A)) be the collection of subages of A which are ages of almost multichainable relational structures. Ordered by set inclusion Id(Almulti(A)) is well founded. Let Id(Almulti <α (A)) be the collection of those which have height < α. According to Theorem 5.53 Id(Almulti <ω 2 (A)) is level finite Problem 16. Does Id(Almulti <ω 2 (A)) always level finite (for α ≤ ω ω )? Theorem 5.54. If an age A contains ages of height ω ⋅ n, for each integer n, then it it contains an age of height ω 2 .Proof. According to Theorem 5.53, Id(Almulti <ω 2 (A)) is level-finite. Hence, by König's lemma, Id(Almulti <ω 2 (A)) contains a chain of order type at least ω 2 .

	Our last result refines Proposition 5.49.

  Proof. First, let A be a subset of N and R A ∶= (N, (ρ m ) m∈A ) where ρ m ∶= {(0, . . . , m)} for m ∈ A. Then φ R A (n) = |A ∩ {0, . . . , n -1}| + 1 for every n ∈ N. Indeed, observe that if F and F ′ are two n-element subsets of N, then the restrictions R A↾F and R A↾F ′ are isomorphic if and only if the largest a, a ′ ∈ A such that {0, . . . , a} ⊆ F and {0, . . . , a ′ } ⊆ F ′ are equal. Next, let ψ ∶ N → N be any non-decreasing and unbounded map such that 1

5. (cf.

[START_REF] Pouzet | Application de la notion de relation presque-enchaînable au dénombrement des restrictions finies d'une relation[END_REF]

, l.7, p. 318 and

[START_REF] Pouzet | The profile of relations[END_REF]

, Theorem 2.10). For every non-decreasing and unbounded map φ ∶ N → N, there is a relational structure R such that φ R is unbounded and eventually bounded above by φ.

This notation is to avoid confusion with the β derivative of I(P ).

We have surveyed several results and problems concerning wqo classes closed under induced substructures. We are forced to note that basic questions asked in the seventies are still unsolved. At least, our knowledge has progressed on some particular classes, like bichains. This could suggest putting some emphasis on ordered structures and particularly ordered graphs. Also, several interesting classes are closed under substructures (that is, via injective homomorphisms). Those made of graphs which are wqo have been characterized by Ding [58]. Does this extend to relational structures?

exactly the local isomorphisms of R ↾W , which can be extended by the identity on F to local isomorphisms of R.

We note that the finiteness of F relies on the fact that the signature of R is finite. As shown in [START_REF] Fraïssé | L'intervalle en théorie des relations, ses généralisations, filtre intervallaires et clôture d'une relation[END_REF], the notion of finite-val set gives rise to a Boolean structure.

Theorem 5.47. The set of val-finite subsets of the domain V of a relational structure with a finite signature forms a Boolean algebra: it contains V , ∅ and it is closed under complemention, finite unions and intersections.

The following fact will be used in the proof of Proposition 5.49.

Lemma 5.48. Let R be a relational structure, K be a subset of V (R) such that R is K-multichainable with parameters W and L ∶= (D, ≤). If m is any integer, the number of m-equivalence classes over K is bounded by an integer depending on w ∶= |W | and m.

Proof. We may suppose that V (R) = K ∪(D×W ) and that for every local isomorphism f of the chain L the map (f, 1 W ). extended by the identity on K is a local isomorphism of R. In this case, θ R,K (m) ≤ (w×m) m .

The first occurrence of the notion of a val-finite set in this context is given by the following. Proposition 5.49. Let A be the age of a relational structure with finite signature. Then A contains a chain of subages of order type ω ⋅ n for every integer n whenever the complement of the kernel ker(R) of some R with age A is not val-finite over ker(R).

Proof. The proof will use Theorem 3.37. With this result, it suffices to prove that if the complement of the kernel of some relational structure R with Age(R) = A is not val-finite, then for each integer n, the set Id(A) of subages of A contains a chain (A p ) 1≤p≤n of subages such that each interval [A p , A p+1 ], 1 ≤ p < n contains infinitely many subages. Indeed, the collection of chains of type ω +1 in [A p , A p+1 ], p < n, provides the required chain of type ω ⋅ n.

We will prove that A contains the age of an almost multichainable relational structure S such that H(Age(S)) = ω ⋅ n + r for some integer r. The domain of the structure S will be of the form V (S) = K ∪ (D × W ), where K = ker S and S ↾W and L ∶= (D, ≤) are the parameters, and such that W contains a strictly increasing sequence W 1 ⊆, . . . , ⊆ W n = W , with

Let V p ∶= K ∪ (L × W p ) and S p ∶= S ↾Vp for p ≤ n. Provided that the ages A p ∶= Age(S p ) are all distinct, the intervals [A p , A p+1 ], p < n will contain infinitely many subages. If not, then, according to Formula 3.2 of Subsection 3.6.2, ker(S p+1 ) = ker(S p ) ∪ F ′ , for some finite subset F ′ of V p+1 . But all S p 's have the same kernel, namely, K, and hence every finite subset of V p+1 can be embedded in V p , thus proving that A p = A p+1 . To ensure that the A p are distinct, we choose some R with Age(R) = A such the complement of the kernel A of R is not val-finite.

Let k be a positive integer such that R contains infinitely many k-equivalence classes over A. Define (W p ) p≤n as follows. Pick a k-tuple (b 1 , . . . , b k ) in V ∖ A and set W 0 ∶= {b 1 , . . . , b k }, and suppose that W p was defined. According to Lemma 5.48, the number of k-equivalence classes over A of any extension multichainable over A of the form L × W p is bounded. Since the number of k-equivalence classes over A is infinite, some k-tuple {b ′ 1 , . . . , b ′ k } of V ∖ A is not equivalent to any of the k-tuples extracted from the above extension. Set W p+1 ∶= W p ∪ {b ′ 1 , . . . , b ′ k }. Now, W being defined, if for every finite subset F of A we can find infinitely many copies of W pairwise disjoint, and disjoint from V ∖ A, then we will be able to extract a family of these copies such that the restriction of R to the union U of these copies augmented with F , forms a multiple of W by an infinite chain which is F -multichainable. Choose F such that the local isomorphisms of R ↾U that extend the identity on A, are exactly those extensions of the identity on F . Then F is the kernel of R ↾U . Hence, setting S ∶= R ↾U , we get the required conclusion. According to Proposition 5.4.2, from R as above we may find an extension R ′ with the same age such that R ′ ↾V ′ ∖A is a multiple of R ↾V ∖A over A by a chain L. Hence, if we contruct W in R ′ instead of R, we will be able to extract infinitely many copies mentioned above, which completes 7.2. The number of wqo ages of graphs. To a word µ let us associate the graph G µ whose vertex set V (G µ ) is {-1, 0, . . . , n -1} if the domain of µ is {0, . . . , n -1}, {-1} ∪ N if the domain of µ is N, and N * or Z if the domain of µ is N * or Z. For two vertices i, j with i < j we let {i, j} be an edge of G µ if and only if µ j = 1 and j = i + 1, or µ j = 0 and j ≠ i + 1.

For instance, if µ is the word defined on N by setting µ i = 1 for all i ∈ N, then G µ is the infinite one way path on {-1} ∪ N. Note that if µ ′ is the word defined on N by setting µ ′ i = 1 for all i ∈ N ∖ {1} and µ ′ 1 = 0, then G µ ′ is also the infinite one way path. In particular, the graphs G µ and G µ ′ have the same age but µ and µ ′ do not have the same sets of finite factors.

0-1 words of length two and their corresponding graphs.

This correspondence between 0-1 words and graphs was considered in [START_REF] Sobrani | Structure d'ordre de la collection des âges de relations[END_REF], (cf. Chapitre II, "Une classification des ages par la hauteur", page [START_REF] Bonnet | Twin-width IV:ordered graphs and matrices[END_REF], and also in [START_REF] Sobrani | Sur les âges de relations et quelques aspects homologiques des constructions D+M[END_REF].

The notion of primality plays a crucial role in the study of the graphs associated to words. We first note that all 0-1-graphs G µ on N, except four, are prime.

Proposition 7.5. If µ is a word with domain N then G µ is prime if and only if µ / ∈ {011111 . . . , 100000 . . . , 0011111 . . . , 1100000 . . .}. In particular, if µ is a recurrent then G µ is prime.

An abstract characterization is the following:

For a word µ, let l(µ) be the supremum of the length of factors of µ made of 1 or of 0. Note that l(µ) is finite if µ is uniformly recurrent and nonconstant.

I list below some properties we will use, see [START_REF] Oudrar | Minimal prime ages, words and permutation graphs[END_REF]: In general, it is not true that two words with different sets of finite factors give different ages of graphs. But, as proved in [START_REF] Oudrar | Minimal prime ages, words and permutation graphs[END_REF]: Theorem 7.8. Let µ and µ ′ be two words. If µ is recurrent and Age(G µ ) ⊆ Age(G µ ′ ), then Fac(µ) ⊆ Fac(µ ′ ).

Proof. Let u ∈ Fac(µ). We prove that u ∈ Fac(µ ′ ). Apply Item (c) of Lemma 7.7 and get some v ∈ {0, 1} * with |v| ≥ 4 such that vu ∈ Fac(µ) and G vu is prime. Since G vu ∈ Age(G µ ) ⊆ Age G µ ′ , G vu embeds in G w for some w ∈ µ ′ . If l(µ) < 4 then it follows from Item (d) that u is a factor of w. If l(µ) ≥ 4 then there is u ′ ∈ Fac(µ) such that u is a factor of u ′ and either 0 4 u ′ or 1 4 u ′ is a factor of µ. It follows from Item (e) that u ′ is a factor of w, and so is u. Hence, u ∈ Fac(µ ′ ).

With the fact that there are 2 ℵ 0 0-1 recurrent words with distinct sets of factors, one obtains that there are 2 ℵ 0 ages of the form Age(G µ ). Those ages are not necessarily wqo. To obtain wqo ages, it suffices to consider graphs associated to uniformly recurrent sequences.

This correspondence above between words and graphs appeared also in [START_REF] Zverovich | Extension of hereditary classes with substitutions[END_REF] and then in [START_REF] Chudnovsky | Unavoidable induced subgraphs in large graphs with no homogeneous sets[END_REF], the resulting graphs being called chain graphs. It was pointed out to us by Brignall [START_REF] Brignall | [END_REF] that chain graphs are the same objects as pin sequences (see [START_REF] Brignall | Simplicity in relational structures and its application to permutation classes[END_REF] Subsection 2.6. p.41). In [START_REF] Oudrar | Minimal prime ages, words and permutation graphs[END_REF] the following was proved: Theorem 7.9. For every 0-1 word µ the age Age(G µ ) consists of permutation graphs.

By considering uniformly recurrent words, one gets that Corollary 7.10. There are 2 ℵ 0 distinct wqo ages, each consisting of permutation graphs.

The ages that we have constructed are not only distinct but they are in fact incomparable under set inclusion. They fit in a category presented in the next subsection. 7.3. Primality and minimality. The notions of primality and minimality are linked in two ways. Definition 7.11. A binary relational structure R is minimal prime if R is prime and R embeds in every induced prime substructure with the same cardinality. Definition 7.12. A hereditary class C made of binary structures is minimal prime if it contains infinitely many prime structures, while every proper hereditary subclass contains only finitely many prime structures.

The first notion appears in [START_REF] Pouzet | On minimal prime graphs and posets[END_REF], the second notion appears in the thesis of Oudrar [START_REF] Oudrar | Sur l'énumération de structures discrètes: une approche par la théorie des relations[END_REF] (see Theorem 5.12, p. 96, and Theorem 5.15, p. 98 of [START_REF] Oudrar | Sur l'énumération de structures discrètes: une approche par la théorie des relations[END_REF]) Several examples of minimal prime graphs and posets are given in [START_REF] Pouzet | On minimal prime graphs and posets[END_REF]. We give below several properties of minimal prime structures and minimal prime classes. A full description of minimal prime hereditary classes of graphs is given in Subsection 7.4.

The relevance of the notion of minimal prime classes with the counting of wqo hereditary classes discussed in the previous section is due to the fact these classes are wqo (Theorem 7.14) and that due to their definition these classes are pairwise incomparable with respect to set inclusion.

We begin with some general properties of minimal prime hereditary classes. Denote by P rim µ the class of finite binary structures of signature µ which are prime. Set P rim(C) ∶= P rim µ ∩ C for every C ⊆ Ω µ . Say that a subclass D of P rim µ is hereditary if it contains every member of P rim µ that can be embedded into some member of D.

These minimal prime classes satisfy the following properties (cf. Théorème 5.14, p.97, [START_REF] Oudrar | Sur l'énumération de structures discrètes: une approche par la théorie des relations[END_REF]).

Theorem 7.13. A hereditary class C of Ω µ is minimal prime if and only if P rim(C) is a Jónsson poset which is cofinal in C.

Theorem 7.14. A hereditary class of finite binary structures which contains infinitely many prime structures contains a minimal prime hereditary subclass, provided that the structures have a finite signature.

Theorem 7.13 is a translation of the definition of minimal prime class. However, in Theorem 7.14, a condition, like the one on the signature, is needed. A proof of Theorem 7.14 relies on the following lemma.

We deduce: Corollary 7.18. Let R ∶= (V, (ρ i ) i∈I ) be a relational structure made of finitely many binary relations and let a ∈ V . If Age(R ↾V ∖{a} ) is hereditarily well-quasi-ordered, then Age(R) is hereditarily well-quasi-ordered.

Proof. If a / ∈ ker(R), there is nothing to prove. If a ∈ ker(R), we set C ∶= Age(R ↾V ∖{a} ). We observe that Age(R) ⊆ C + and we apply Lemma 7.17.

Proof of the third part of Theorem 7.16. Let C be a minimal prime hereditary class and R be such that Age(R) = C. Suppose that ker(R) is nonempty. Let a ∈ ker(R). Then Age(R ↾V ∖{a} ) / = Age(R) = C. Since C is minimal prime, Age(R ↾V ∖{a} ) contains only finitely many primes. Theorem 4. [START_REF] Berstel | Noncommutative rational series with applications[END_REF] As mentioned in Subsection 3.4, it is not known if a hereditary class of finite graphs which is wqo is bqo.

Problem 20.

(1) Is every minimal prime hereditary class of finite binary structures bqo?

(2) Is it true that | ker(R)| ≤ 2 if Age(R) minimal prime? (3) Is the number of exhaustible minimal prime ages finite? (4) Is it true that the age of a minimal prime binary structure is necessarily minimal prime? As we will see, the answers to (2), (3), ( 4) are positive if one considers minimal prime classes of graphs. In this case, there are only five examples with a nonempty kernel. Even in the case of graphs we do not know the answer to (4). The converse is false in the sense that there are minimal prime ages of graphs such that no graph with that age is minimal prime.

We prove:

Theorem 7.20. If C is minimal prime and exhaustible, then every binary prime structure R with Age(R) = C embeds a minimal prime structure.

The proof relies on a result of Delhommé and McKay, Theorem 8.2, presented in Section 8 and Lemma 7.17. We first prove the following result. Lemma 7.21. If C is minimal prime and exhaustible then C ≤ω is well-quasi-ordered.

). This age contains only finitely many primes. From Theorem 8.2, D ≤ω is well-quasi-ordered. Furthermore, members of D ≤ω when labelled by any finite set form a well-quasi-ordered set. According to Lemma 7.17, (D ≤ω ) +1 has the same property. Next, C ≤ω ⊆ (D ≤ω ) +1 . Indeed, every member of C ≤ω has a copy R ′ in a countable extension R ′′ of R having the same age as R, therefore, Age(R ′ ↾V (R ′ )∖{a} ) ⊆ C. Hence, C ≤ω is well-quasi-ordered. We now prove the following result. Lemma 7.22. Let C be a hereditary class of Ω µ . If C ≤ω is well-founded then every prime member of C ≤ω , if any, embeds a minimal one.

Proof of Theorem 7.20. Let R be a prime structure with Age(R) = C. According to Lemma 7.21, C ≤ω is well-quasi-ordered. According to Lemma 7.22, R embeds a minimal prime member. ◻ 7.4. Minimal prime ages of graphs. We now give a complete description of minimal prime ages of graphs.

It is based on a description by Chudnovsky et al. [START_REF] Chudnovsky | Unavoidable induced subgraphs in large graphs with no homogeneous sets[END_REF] of unavoidable prime graphs in large finite prime graphs (see also Malliaris and Terry [130]). Our result is based on our study of graphs associated to 0-1 sequences.

Theorem 7.23. Let µ be a 0-1 sequence on N. The following statements are equivalent.

(i) µ is uniformly recurrent.

(ii) µ is recurrent and Age(G µ ) is minimal prime.

Proof (ii) ⇒ (i). If µ is constant, µ is uniformly periodic. So, we may suppose that µ is nonconstant. In this case, l(µ) is bounded. Otherwise, Fac(0) or Fac(1) (where 0 and 1 denote the constant words on N) is included in Fac(µ), hence Age(G 0 ) or Age(G 1 ) is included in Age(G µ ). Since Age(G µ ) is minimal prime, it is equal to one of these ages, amounting to the fact that µ is constant. This is a contradiction.

We now prove that Fac(µ) is Jónsson and we apply Theorem 7.3. First, Fac(µ) is infinite since µ is recurrent. Next, let X be an infinite initial segment of Fac(µ). We prove that X = Fac(µ). Since l(µ) is finite, we may apply Item (b) of Lemma 7.7. It yields that the set X

According to Item (c) of Lemma 7.7, since µ is recurrent and u ∈ Fac(µ) there is some v ∈ {0, 1} * with |v| ≥ 3 such that vu ∈ Fac(µ) and G vu is prime. Since G vu ∈ Age(G µ ) ⊆↓ G X , G vu embeds in G w for some w ∈ X. If l(µ) < 4, it follows from (d) of Lemma 7.7, that u is a factor of w. If l(µ) ≥ 4, then there is u ′ ∈ Fac(µ) such that u is a factor of u ′ and either 0 4 u ′ or 1 4 u ′ is a factor of µ. It follows from Item (e) of Lemma 7.7 applied to either v = 0 4 or v = 1 4 , that u ′ is a factor of w, and so is u. Hence, u ∈ X as required.

There are other minimal prime ages of graphs than those described in Theorem 7.23 above. But there are only eleven such ages.

Let M be the graphs G 0 , G 1 , G 3 , G 4 , G 5 and G 6 depicted in Figures 4 and5. Let M be the list of these graphs and their complements. Let L be the set of the ages of these graphs and of their complements. It should be noted that the graphs G 5 , G comp Proof. An inspection of the six members of M shows that G 0 , G 1 and G 4 are multichainable with an empty kernel, the three others are almost multichainable with a one-element kernel, in the case of G 3 and G 5 , and a two-element kernel in the case of G 6 . This gives three exhaustible ages; with the ages of their complements added (and since G 5 and its complement G comp 5 have the same age) this gives five exhaustible ages. The fact that these graphs are minimal prime is given in [START_REF] Pouzet | On minimal prime graphs and posets[END_REF]. The second part of the theorem, notably the fact that the ages are distinct and minimal prime is detailed in Oudrar's thesis [START_REF] Oudrar | Sur l'énumération de structures discrètes: une approche par la théorie des relations[END_REF], Chapter 6, p. 109.

We conclude with the following result: