Samuele Giraudo
email: samuele.giraudo@u-pem.fr

C Ligm

OPERADS OF DECORATED CLIQUES I: CONSTRUCTION AND QUOTIENTS

A

. We introduce a functorial construction C which takes unitary magmas as input and produces operads. The obtained operads involve configurations of chords labeled by elements of , called -decorated cliques and generalizing usual configurations of chords. By considering combinatorial subfamilies of -decorated cliques defined, for instance, by limiting the maximal number of crossing diagonals or the maximal degree of the vertices, we obtain suboperads and quotients of C . This leads to a new hierarchy of operads containing, among others, operads on noncrossing configurations, Motzkin configurations, forests, dissections of polygons, and involutions. Besides, the construction C leads to alternative definitions of the operads of simple and double multi-tildes, and of the gravity operad.

Up to some restrictions or enrichments, sets of these objects can be put in bijection with several combinatorial families. For instance, it is well-known that triangulations [START_REF] De Loera | Triangulations[END_REF], forming a particular subset of the set of all configurations of chords, are in one-to-one correspondence with binary trees, and a lot of structures and operations on binary trees translate nicely on triangulations. Indeed, among others, the rotation operation on binary trees [Knu98] is the covering relation of the Tamari order [START_REF] Huang | Problems of associativity: A simple proof for the lattice property of systems ordered by a semi-associative law[END_REF] and this operation translates as a diagonal flip in triangulations. Also, noncrossing configurations [START_REF] Flajolet | Analytic combinatorics of non-crossing configurations[END_REF] form another interesting subfamily of such chord configurations. Natural generalizations of noncrossing configurations consist in allowing, with more or less restrictions, some crossing diagonals. One of these families is formed by the multi-triangulations [CP92] wherein the number of mutually crossing diagonals is bounded. In particular, the class of combinatorial objects in bijection with some configurations of chords is large enough in order to contain, among others, dissections of polygons, noncrossing partitions, permutations, and involutions.

On the other hand, coming historically from algebraic topology [START_REF] May | The geometry of iterated loop spaces[END_REF][START_REF] Boardman | Homotopy invariant algebraic structures on topological spaces[END_REF], operads provide an abstraction of the notion of operators (of any arities) and their compositions. In more concrete terms, operads are algebraic structures abstracting the notion of planar rooted trees and their grafting operations (see [START_REF] Loday | Algebraic Operads[END_REF] for a complete exposition of the theory and [Mén15] for an exposition focused on symmetric set-operads). The modern treatment of operads in algebraic combinatorics consists in regarding combinatorial objects like operators endowed with gluing operations mimicking the composition of operators. In the last years, a lot of combinatorial sets and combinatorial spaces have been endowed fruitfully with the structure of an operad (see for instance [Cha08] for an exposition of known interactions between operads and combinatorics, focused on trees, [LMN13, GLMN16] where operads abstracting operations in language theory are introduced, [CG14] for the study of an operad involving particular noncrossing configurations, [START_REF] Giraudo | Combinatorial operads from monoids[END_REF] for a general construction of operads on many combinatorial sets, [Gir16a] where operads are constructed from posets, and [CHN16] where operads on various species of trees are introduced). In most of the cases, this approach brings results about enumeration, helps to discover new statistics, and leads to establish new links (by morphisms) between different combinatorial sets or spaces. We can observe that most of the subfamilies of polygons endowed with configurations of chords discussed above are stable for several natural composition operations. Even better, some of these can be described as the closure with respect to these composition operations of small sets of polygons. For this reason, operads are very promising candidates, among the modern algebraic structures, to study such objects under an algebraic and combinatorial flavor.

The purpose of this work is twofold. First, we are concerned with endowing the linear span of the configurations of chords with the structure of an operad. This leads to seeing these objects under a new light, stressing some of their combinatorial and algebraic properties. Second, we would provide a general construction of operads of configurations of chords rich enough so that it includes some already known operads. As a consequence, we obtain alternative definitions of existing operads and new interpretations of these. For this aim, we work here with -decorated cliques (or -cliques for short), that are complete graphs whose arcs are labeled by elements of , where is a unitary magma. These objects are natural generalizations of configurations of chords since the arcs of any -clique labeled by the unit of are considered as missing. The elements of different from the unit allow moreover to handle chords of different colors. For instance, each usual noncrossing configuration c can be encoded by an N 2 -clique p, where N 2 is the cyclic additive unitary magma Z/2Z, wherein each arc labeled by 1 ∈ N 2 in p denotes the presence of the same arc in c, and each arc labeled by 0 ∈ N 2 in p denotes its absence in c. Our construction is materialized by a functor C from the category of unitary magmas to the category of operads. It builds, from any unitary magma , an operad C on -cliques. The partial composition p • q of two -cliques p and q of C consists in gluing the th edge of p (with respect to a precise indexation) and a special arc of q, called the base, together to form a new -clique. The magmatic operation of explains how to relabel the two overlapping arcs.

This operad C has a lot of properties, which can be apprehended both under a combinatorial and an algebraic point of view. First, many families of particular configurations of chords form quotients or suboperads of C . We can for instance control the degrees of the vertices or the crossings between diagonals to obtain new operads. We can also forbid all diagonals, or some labels for the diagonals or the edges, or all nestings of diagonals, or even all cycles formed by arcs. All these combinatorial particularities and restrictions on -cliques behave well algebraically. Moreover, by using the fact that the direct sum of two ideals of an operad is still an ideal of , these constructions can be mixed to get even more operads. For instance, it is well-known that Motzkin configurations, that are polygons with disjoint noncrossing diagonals, are enumerated by Motzkin numbers [START_REF] Th | Relations between hypersurface cross ratios, and a combinatorial formula for partitions of a polyion, for permanent preponderance, and for non-associative products[END_REF]. Since a Motzkin configuration can be encoded by an -clique where all vertices are of degree at most 1 and no diagonal crosses another one, we obtain an operad Mot on colored Motzkin configurations which is both a quotient of Deg 1 , the quotient of C consisting in all -cliques such that all vertices are of degree at most 1, and of Cro 0 , the quotient (and suboperad) of C consisting in all noncrossing -cliques. We also get quotients of C involving, among others, Schröder trees, forests of paths, forests of trees, dissections of polygons, Lucas configurations, with colored versions for each of these. This leads to a new hierarchy of operads, wherein links between its components appear as surjective or injective operad morphisms. One of the most notable of these is built by considering the D 0 -cliques that have vertices of degree at most 1, where D 0 is the multiplicative unitary magma on {0 1}. This is in fact the quotient Deg 1 D 0 of CD 0 and involves involutions (or equivalently, standard Young tableaux by the Robinson-Schensted correspondence [START_REF] Lothaire | Algebraic combinatorics on words. Encyclopedia of mathematics and its applications[END_REF]). To the best of our knowledge, Deg 1 D 0 is the first nontrivial operad on these objects.

As an important remark at this stage, let us highlight that when is nontrivial, C is not a binary operad. Indeed, all its minimal generating sets are infinite and its generators have arbitrarily high arities. Furthermore, the construction C maintains some links with the operad RatFct of rational functions introduced by Loday [START_REF] Loday | On the operad of associative algebras with derivation[END_REF]. In fact, provided that satisfies some conditions, each -clique encodes a rational function. This defines an operad morphism from C to RatFct. Moreover, the construction C allows to construct already known operads in original ways. For instance, for well-chosen unitary magmas , the operads C contain MT and DMT, two operads respectively defined in [LMN13] and [GLMN16] that involve multi-tildes and double multi-tildes, operators coming from formal language theory [CCM11]. The operads C also contains Grav, the gravity operad, a symmetric operad introduced by Getzler [START_REF] Getzler | Two-dimensional topological gravity and equivariant cohomology[END_REF], seen here as a nonsymmetric one [START_REF] Alm | Brown's dihedral moduli space and freedom of the gravity operad[END_REF].

This text is organized as follows. Section 1 sets our notations, general definitions, and tools about nonsymmetric operads (since we deal only with nonsymmetric operads here, we call these simply operads) and configurations of chords. In Section 2, we introduce -cliques, the construction C, and study some of its properties. Then, Section 3 is devoted to define several suboperads and quotients of C . This leads to plenty of new operads on particular -cliques. Finally, in Section 4, we use the construction C to provide alternative definitions of some known operads.

This paper is an extended version of [START_REF] Giraudo | Combalgebraic structures on decorated cliques[END_REF], containing the proofs of the presented results.

Acknowledgements. The author would like to thank warmly Dan Petersen for introducing him to the gravity operad and highlighting links between this operad and the current work. The author also thanks the anonymous reviewer for his time and his suggestions, which have greatly contributed to improving the article.

E

We set here our notations and recall some definitions about operads and related structures. We also introduce some notations and definitions about configurations of chords in polygons.

1.1. Nonsymmetric operads. We adopt most of notations and conventions of [START_REF] Loday | Algebraic Operads[END_REF] about operads. For the sake of completeness, we recall here the elementary notions about operads employed thereafter.

A nonsymmetric operad in the category of vector spaces, or a nonsymmetric operad for short, is a graded vector space

:= 1 ( ) (1.1.1)
together with linear maps

• : ( ) ⊗ ( ) → ( + -1) 1 ∈ [ ] (1.1.2)
called partial compositions, and a distinguished element 1 ∈ (1), the unit of . This data has to satisfy the three relations

( • ) • + -1 = • ( • ) ∈ ( ) ∈ ( ) ∈ ( ) ∈ [ ] ∈ [ ] (1.1.3a) ( • ) • + -1 = ( • ) • ∈ ( ) ∈ ( ) ∈ ( ) < ∈ [ ] (1.1.3b) 1 • 1 = = • 1 ∈ ( ) ∈ [ ] (1.1.3c)
Since we consider in this paper only nonsymmetric operads, we shall call these simply operads. Moreover, in this work, we shall only consider operads for which (1) has dimension 1.

When is such that all ( ) have finite dimensions for all 1, the Hilbert series of is the series ( ) defined by

( ) := 1 dim ( ) (1.1.4)
If is an element of such that ∈ ( ) for an 1, we say that is the arity of and we denote it by | |. If 1 and 2 are two operads, a linear map φ : 1 → 2 is an operad morphism if it respects arities, sends the unit of 1 to the unit of 2 , and commutes with partial composition maps. We say that 2 is a suboperad of 1 if 2 is a graded subspace of 1 , 1 and 2 have the same unit, and the partial compositions of 2 are the ones of 1 restricted on 2 . For any subset G of , the operad generated by G is the smallest suboperad G of containing G. When G = and G is minimal with respect to the inclusion among the subsets of G satisfying this property, G is a minimal generating set of and its elements are generators of . An operad ideal of is a graded subspace I of such that, for any ∈ and ∈ I, • and • are in I for all valid integers and . Given an operad ideal I of , one can define the quotient operad /I of by I in the usual way.

Let us recall and set some more definitions about operads. The Hadamard product between the two operads 1 and 2 is the operad

1 * 2 satisfying ( 1 * 2 )( ) = 1 ( )⊗ 2 (
), and its partial composition is defined component-wise from the partial compositions of 1 and

2 . An element of (2) is associative if

• 1 = • 2 .
An antiautomorphism of is a graded vector space automorphism φ of sending the unit of to the unit of and such that for any ∈ ( ), ∈ , and satisfying, for all ∈ ( ), ∈ ( ), and

∈ [ ], φ( • ) = φ( ) • -+1 φ(
∈ [ ], ρ(1) = 1 (1.1.8a) ρ +1 ( ) = (1.1.8b) ρ( • ) = ρ( ) • ρ( ) if = 1 ρ( ) • -1 otherwise (1.1.8c)
We call such a map ρ a rotation map. ) different from ( + 1) and (1 + 1), and an edge is an arc of the form ( + 1) and different from (1 + 1). We denote by p (resp.

p , p ) the set of all arcs (resp. diagonals, edges) of p. For any ∈ [ ], the th edge of p is the edge ( + 1), and the arc (1 + 1) is the base of p.

In our graphical representations, each polygon is drawn so that its base is the bottommost segment, vertices are implicitly numbered from 1 to + 1 in clockwise direction, and the diagonals are not drawn. For example,

p := 1 2 3 4 5 6 (1.2.1)
is a polygon of size 5. Its set of all diagonals is )). When the base of p is labeled, we write simply p 0 for p(1 +1), where is the size of p. Finally, when the th edge of p is labeled, we write simply p for p( + 1).

p = {(1 3) (1 4) (1 5) (2 4) (2 5) (2 6) (3 5) (3 6) (4 6)} (1.2.2) its set of all edges is p = {(1 2) (2 3) (3 4) ( 4
In our graphical representations, we shall represent any S-configuration p by drawing a polygon of the same size as the one of p following the conventions explained before, and by labeling its arcs accordingly. For instance

p := a b a b (1.2.6)
is an {a b}-configuration. The arcs (1 2) and (1 4) of p are labeled by a, the arcs (2 5) and (4 5) are labeled by b, and the other arcs are unlabeled.

Additional definitions.

Let us now provide some definitions and statistics on configurations. Let p be a configuration of size . The skeleton of p is the undirected graph skel(p) on the set of vertices [ + 1] such that for any < ∈ [ + 1], there is an arc { } in skel(p) if ( ) is labeled in p. The degree of a vertex of p is the number of vertices adjacent to in skel(p). The degree degr(p) of p is the maximal degree among its vertices.

F

We describe in this section our construction from unitary magmas to operads and study its main algebraic and combinatorial properties.

2.1. Operads of decorated cliques. We present here our main combinatorial objects, the decorated cliques. The construction C, which takes a unitary magma as input and produces an operad, is defined.

2.1.1. Unitary magmas. Recall first that a unitary magma is a set endowed with a binary operation admitting a left and right unit 1 . For convenience, we denote by ¯ the set \ {1 }. To explore some examples in this article, we shall mostly consider four sorts of unitary magmas: the additive unitary magma on all integers denoted by Z, the cyclic additive unitary magma on Z/ Z denoted by N , the unitary magma

D := {1 0 d 1 d } (2.1.1)
where ) of p is labeled by an element different from 1 , we say that the arc ( ) is solid. By convention, we require that the -clique of size 1 having its base labeled by 1 is the only such object of size 1. The set of all -cliques is denoted by .

In our graphical representations, we shall represent any -clique p by following the drawing conventions of configurations explained in Section 1.2.2 with the difference that non-solid diagonals are not drawn. For instance,

p := -1 2 1 -1 3 2 1 (2.1.4)
is a Z-clique such that, among others p(1 2) = -1, p(1 5) = 2, p(3 7) = -1, p(5 7) = 1, p(2 3) = 0 (because 0 is the unit of Z), and p(2 6) = 0 (for the same reason).

Let us now provide some definitions and statistics on -cliques. The underlying configuration of p is the ¯ -configuration p of the same size as the one of p and such that p( ) := p( ) for all solid arcs ( ) of p, and all other arcs of p are unlabeled. The skeleton, (resp. degree, crossing number) of p is the skeleton (resp. the degree, the crossing number) of p. Moreover, p is nesting-free, (resp. acyclic, white, an -bubble, an -triangle), if p is nesting-free (resp. acyclic, white, a bubble, a triangle). The set of all -bubbles (resp.

-triangles) is denoted by (resp. ).

2.1.3. Partial composition of -cliques. From now, the arity of an -clique p is its size and is denoted by |p|. For any unitary magma , we define the vector space

C := 1 C ( ) = K (2.1.5)
where C ( ) is the linear span of all -cliques of arity , 1. The set forms hence a basis of C called fundamental basis. Observe that the space C (1) has dimension 1 since it is the linear span of the -clique . We endow C with partial composition maps

• : C ( ) ⊗ C ( ) → C ( + -1) 1 ∈ [ ] (2.1.6)
defined linearly, in the fundamental basis, in the following way. Let p and q be two cliques of respective arities and , and ∈ [ ] be an integer. We set p• q as the -clique of arity + -1 such that, for any arc ( ) where 1

< + , (p • q)( ) :=                        p( ) if p( -+ 1) if < + and ( ) = ( + ) p( -+ 1 -+ 1) if + q( -+ 1 -+ 1) if < + and ( ) = ( + ) p q 0 if ( ) = ( + ) 1 otherwise (2.1.7)
We recall that denotes the operation of and 1 its unit. Graphically, p • q is obtained by gluing the base of q onto the th edge of p and by labeling this arc by p q 0 , and by adding all required non solid diagonals on the graph thus obtained to become a clique (see Figure 1). For example, in CZ, one has the two partial compositions

p +1 p • q 0 q = p +1 p q 0 q = + p q 0 F
. The partial composition of C , described in graphical terms. Here, p and q are two -cliques. The arity of q is and is an integer between 1 and |p|.

1 -2 -2 1 • 2 1 3 1 2 = 1 -2 1 1 1 2 1 (2.1.8a) 1 -2 -2 1 • 2 1 2 1 2 = 1 -2 1 1 2 1 (2.1.8b)
2.1.4. Functorial construction from unitary magmas to operads. If 1 and 2 are two unitary magmas and θ :

1 → 2 is a unitary magma morphism, we define

Cθ : C 1 → C 2 (2.1.9)
as the linear map sending any 1 -clique p of arity to the 2 -clique (Cθ)(p) of the same arity such that, for any arc ( ) where 1

< + 1, ((Cθ)(p))( ) := θ(p( )) (2.1.10)
Graphically, (Cθ)(p) is the 2 -clique obtained by relabeling each arc of p by the image of its label by θ.

Theorem 2.1.1. The construction C is a functor from the category of unitary magmas to the category of operads. Moreover, C respects injections and surjections.

Proof. Let be a unitary magma. The fact that C endowed with the partial composition (2.1.7) is an operad can be established by showing that the two associativity relations (1.1.3a) and (1.1.3b) of operads are satisfied. This is a technical but a simple verification. Since C (1) contains and this element is the unit for this partial composition, (1.1.3c) holds. Moreover, let 1 and 2 be two unitary magmas and θ : 1 → 2 be a unitary magma morphism. The fact that the map Cθ defined in (2.1.10) is an operad morphism is straightforward to check. All this implies that C is a functor. Finally, the fact that C respects injections and surjections is also straightforward to verify.

We name the construction C as the clique construction and C as the -clique operad. Observe that the fundamental basis of C is a set-operad basis of C . Besides, when is the trivial unitary magma {1 }, C is the linear span of all decorated cliques having only non-solid arcs. Thus, each space C ( ), 1, is of dimension 1 and it follows from the definition of the partial composition of C that this operad is isomorphic to the associative operad As. The next result shows that the clique construction is compatible with the Cartesian product of unitary magmas.

Proposition 2.1.2. Let

1 and 2 be two unitary magmas. Then,

C( 1 × 2 ) is isomorphic to the Hadamard product of operads (C 1 ) * (C 2 ). Proof. Let φ : (C 1 ) * (C 2 ) → C( 1 ×
2 ) be the linear map defined as follows. For any 1 -clique p of C 1 and any 2 -clique q of C 2 both of arity , φ(p ⊗ q) is the 1 × 2 -clique defined, for any 1

< + 1, by (φ(p ⊗ q)) ( ) := (p( ) q( )) (2.1.11) Let the linear map ψ : C( 1 × 2 ) → (C 1 ) * (C 2 ) defined, for any 1 × 2 -clique r of C( 1 ×
2 ) of arity , as follows. The 1 -clique p and the 2 -clique q of arity of the tensor p ⊗ q := ψ(r) are defined, for any 1 < + 1, by p( ) := and q( ) := where ( ) = r( ). Since we observe immediately that ψ is the inverse of φ, φ is a bijection. Moreover, it follows from the definition of the partial composition of clique operads that φ is an operad morphism. The statement of the proposition follows.

2.2. General properties. We investigate here some properties of clique operads, as their dimensions, their minimal generating sets, the fact that they admit a cyclic operad structure, and describe their partial compositions over two alternative bases.

< ∈ [ + 1], R if p( ) = a R if p( ) = b R and R if p( ) = c (2.2.1)
In particular, when = 2 (resp. = 3, = 4), = {1 c} (resp. = {1 a b}, = {1 a b c}) and the set of all -cliques of arities 2 is in one-to-one correspondence with the set of all irreflexive and symmetric (resp. irreflexive and antisymmetric, irreflexive) binary relations on [ + 1]. Therefore, the operads C can be interpreted as operads involving binary relations with more or less properties.

Dimensions and minimal

(i) the diagonal (
) is solid and its crossing number is 0, or ( ) is not solid; (ii) the -clique p can be written as p = q• r, where q is an -clique of arity + -+1 and r is an -clique of arity -.

Proof. Assume first that (i) holds. Set q as the -clique of arity + -+ 1 defined, for any arc ( ) where 1

< + -+ 2, by q( ) :=        p( ) if p( + --1) if + 1 p( + --1 + --1) otherwise (2.2.4)
and r as the -clique of aritydefined, for any arc ( ) where 1

< -+ 1, by r( ) := p( + -1 + -1) if ( ) = (1 -+ 1) 1 otherwise (2.2.5)
By following the definition of the partial composition of C , one obtains p = q • r, hence (ii) holds.

Assume conversely that (ii) holds. By definition of the partial composition of C , the fact that p = q • r implies that p( ) = 1 for any arc ( ) such that ( ) and ( ) are crossing. Therefore, (i) holds.

Let

be the set of all -cliques p of arity 2 that do not satisfy the property of the statement of Lemma 2.2.2. In other words, is the set of all -cliques such that, for any (non-necessarily solid) diagonal ( ) of p, there is at least one solid diagonal ( ) of p such that ( ) and ( ) are crossing. We call the set of all prime -cliques.

Observe that, according to this description, all -triangles are prime.

p 1 q 0 ∈ δ=p 1 q 0 λ p 2 p 1 p 0 λ q 2 q 1 q 0 = 0 p 0 p 2 q 1 q 2 ∈ δ ∈ ¯ (2.2.10a) p 1 q 0 ∈ p 1 q 0 =1 λ p 2 p 1 p 0 λ q 2 q 1 q 0 -λ p 1 q 1 p 0 λ p 2 q 2 q 0 = 0 p 0 p 2 q 1 q 2 ∈ (2.2.10b) p 2 q 0 ∈ δ=p 2 q 0 λ p 2 p 1 p 0 λ q 2 q 1 q 0 = 0 p 0 p 1 q 1 q 2 ∈ δ ∈ ¯ (2.2.10c)
Proof. The element defined in (2.2.9) is associative if and only if

• 1 -• 2 = 0.
Therefore, this property is equivalent to the fact that

• 1 -• 2 =     p q∈ δ:=p 1 q 0 =1 λ p λ q q 2 q 1 p 2 p 0 δ     +     p q∈ p 1 q 0 =1 λ p λ q q 2 q 1 p 2 p 0     -     p q∈ δ:=p 2 q 0 =1 λ p λ q q 1 p 1 q 2 p 0 δ     -     p q∈ p 2 q 0 =1 λ p λ q q 1 p 1 q 2 p 0     =     p 0 p 2 q 1 q 2 ∈ δ∈ ¯     p 1 q 0 ∈ δ=p 1 q 0 λ p 2 p 1 p 0 λ q 2 q 1 q 0     q 2 q 1 p 2 p 0 δ     +     p 0 p 2 q 1 q 2 ∈     p 1 q 0 ∈ p 1 q 0 =1 λ p 2 p 1 p 0 λ q 2 q 1 q 0 -λ p 1 q 1 p 0 λ p 2 q 2 q 0     q 2 q 1 p 2 p 0     -     p 0 p 1 q 1 q 2 ∈ δ∈ ¯     p 2 q 0 ∈ δ=p 2 q 0 λ p 2 p 1 p 0 λ q 2 q 1 q 0     q 1 p 1 q 2 p 0 δ     = 0 (2.2.11)
and hence, is equivalent to the fact that (2.2.10a), (2.2.10b), and (2.2.10c) hold.

For instance, by Proposition 2.2.4, the binary elements

1 1 1 (2.2.12a) + 1 - 1 + 1 -1 1 + 1 1 - 1 1 - 1 1 1 (2.2.12b)
of CN 2 are associative, and the binary elements Proof. Assume first that is right cancelable. Let 1, ∈ [ ], and p, p , and q be three -cliques such that p and p are of arity . If • q (p) = • q (p ), we have p • q = p • q. By definition of the partial composition map of C , we have p( ) = p ( ) for all arcs ( ) where 1 < + 1 and ( ) = ( + 1). Moreover, we have p q 0 = p q 0 Since is right cancelable, this implies that p = p , and hence, p = p . This shows that the maps • q are injective and thus, that the fundamental basis of C is a basic set-operad basis.

0 0 - 0 0 0 (2.2.13a) 0 -0 0 - 0 0 + 0 0 0 (2.2.
Conversely, assume that the fundamental basis of C is a basic set-operad basis. Then, in particular, for all 1 and all -cliques p, p , and q such that p and p are of arity , • q 1 (p) = • q 1 (p ) implies p = p . This is equivalent to the statement that p 1 q 0 = p 1 q 0 implies p 1 = p 1 . This amount exactly to the statement that is right cancelable.

2.2.6. Cyclic operad structure. Let ρ : C → C be the linear map sending any clique p of arity to the -clique ρ(p) of the same arity such that, for any arc ( ) where

1 < + 1, (ρ(p))( ) := p( + 1 + 1) if p(1 + 1) otherwise ( = + 1) (2.2.16)
Graphically, ρ(p) is the -clique obtained by applying a rotation of one step of p in counterclockwise direction. For instance, one has in CZ,

ρ    1 -2 -2 1    = -2 1 1 -2
(2.2.17) Proposition 2.2.7. Let be a unitary magma. The map ρ is a rotation map of C , endowing this operad with a cyclic operad structure.

Proof. The fact that ρ is a rotation map for C follows from a technical but straightforward verification of the fact that Relations (1.1.8a), (1.1.8b), and (1.1.8c) hold.

2.2.7. Alternative bases. If p and q are two -cliques of the same arity, the Hamming distance h(p q) between p and q is the number of arcs (

) such that p( ) = q( ). Let be be the partial order relation on the set of all -cliques, where, for any -cliques p and q, one has p be q if q can be obtained from p by replacing some labels 1 of its edges or its base by other labels of . In the same way, let d be the partial order on the same set where p d q if q can be obtained from p by replacing some labels 1 of its diagonals by other labels of .

For all -cliques p, let us introduce the elements of C defined by

H p := p ∈ p be p p (2.2.18a)
and

K p := p ∈ p d p (-1) h(p p) p (2.2.18b)
For instance, in CZ,

H 1 1 2 2 = 1 2 + 1 2 2 + 1 1 2 + 1 1 2 2 (2.2.19a) K 1 1 2 2 = 1 1 2 2 - 1 2 2 - 1 1 2 + 1 2 (2.2.
H p • H q =              H p• q + H d (p)• q + H p• d 0 (q) + H d (p)• d 0 (q)
if p = 1 and q 0 = 1

H p• q + H d (p)• q if p = 1 and q 0 = 1 H p• q + H p• d 0 (q)
if p = 1 and q 0 = 1

H p• q otherwise (2.2.21)
Proof. From the definition of the H-basis, we have

H p • H q = p q ∈ p be p q be q p • q = p q ∈ p be p q be q p =1 q 0 =1 p • q + p q ∈ p be p q be q p =1 q 0 =1 p • q + p q ∈ p be p q be q p =1 q 0 =1 p • q + p q ∈ p be p q be q p =1 q 0 =1 p • q (2.2.22)
Let 1 (resp. 2 , 3 , 4 ) be the first (resp. second, third, fourth) summand of the right-hand side of (2.2.22). There are four cases to explore depending on whether the th edge of p and the base of q are solid or not. From the definition of the H-basis and of the partial order relation be , we have that (a) when p = 1 and q

0 = 1 , 1 = H p• q , 2 = H p• d 0 (q) , 3 = H d (p)• q , and 4 = H d (p)• d 0 (q) ; (b
) when p = 1 and q 0 = 1 , 1 = 0, 2 = H p• q , 3 = 0, and 4 = H d (p)• q ; (c) when p = 1 and q 0 = 1 , 1 = 0, 2 = 0, 3 = H p• q , and 4 = H p• d 0 (q) ; (d) and when p = 1 and q 0 = 1 , 1 = 0, 2 = 0, 3 = 0, and 4 = H p• q . By assembling these cases together, we obtain the stated result. Proposition 2.2.9. Let be a unitary magma. The partial composition of C can be expressed in terms of the K-basis, for any -cliques p and q different from and any valid integer , by

K p • K q =    K p• q if p q 0 = 1 K p• q + K d (p)• d 0 (q) otherwise (2.2.23)
Proof. Let be the arity of q. From the definition of the K-basis and of the partial order relation d , we have

K p • K q = p q ∈ p d p q d q (-1) h(p p)+h(q q) p • q = p q ∈ p • q d p• q p =p q 0 =q 0 (-1) h(p p)+h(q q) p • q = r∈ r d p• q r( + -1)=p q 0 (-1) h(r p• q) r (2.2.24)
When p q 0 = 1 , (2.2.24) is equal to K p• q . Otherwise, when p q 0 = 1 , we have

r∈ r d p• q r( + -1)=p q 0 (-1) h(r p• q) r = r∈ r d p• q (-1) h(r p• q) r - r∈ r d p• q r( + -1) =p q 0 (-1) h(r p• q) r = K p• q - r∈ r d d (p)• d 0 (q) (-1) h(r p• q) r = K p• q - r∈ r d d (p)• d 0 (q) (-1) 1+h(r d (p)• d 0 (q)) r = K p• q + K d (p)• d 0 (q) (2.2.25)
This proves the claimed formula for the partial composition of C over the K-basis.

For instance, in CZ, ) := 1, 1 is of arity 1 while 2 is of arity 2, so that 1 and 2 are considered as different rational functions. The partial composition of two rational functions ∈ RatFct( ) and ∈ RatFct( ) is defined by

H 1 • 2 H 1 = H + 2 H 1 + H 2 (2.2.26a) K 1 • 2 K 1 = K + K 2 (2.2.26b) H 2 1 • 3 H 1 2 2 = H 2 1 2 + H 2 1 1 2 + H 2 1 2 2 + H 2 1 3 2 (2.2.26c) K 2 1 • 3 K 1 2 2 = K 2 1 2 + K 2 1 3 2 (2.2.26d) H -1 2 1 • 2 H -1 1 1 = H -1 -1 2 1 1 + 2 H -1 2 1 1 + H -1 1 2 1 1 (2.2.26e) K -1 2 1 • 2 K -1 1 1 = K -1 2 1 1 (2.2.26f) and in D 1 , H 0 d 1 0 • 2 H 0 0 0 = 3 H 0 0 d 1 0 0 + H 0 d 1 0 0 (2.2.27a) K 0 d 1 0 • 2 K 0 0 0 = K 0 0 d 1 0 0 + K 0 d 1 0 0 (2.2.
• := ( 1 -1 + • • • + + -1 + + -1 ) ( + -1 ) (2.2.29)
The rational function of RatFct(1) defined by ( 1) := 1 is the unit of RatFct. As shown by Loday, this operad is (nontrivially) isomorphic to the operad Mould introduced by Chapoton [Cha07].

Let us assume that is a Z-graded unitary magma, that is a unitary magma such that there exists a unitary magma morphism θ : → Z. We say that θ is a rank function of . In this context, let

F θ : C → RatFct (2.2.30)
be the linear map defined, for any -clique p, by

F θ (p) := ( )∈ p + • • • + -1 θ(p( ))
(2.2.31)

For instance, by considering the unitary magma Z together with its identity map Id as rank function, one has

F Id     -1 2 1 -2 3 -1     = ( 1 + 2 + 3 + 4 ) 2 ( 1 + 2 + 3 + 4 + 5 + 6 ) 3 4 1 ( 3 + 4 + 5 + 6 ) 2 ( 5 + 6 ) (2.2.32)
Theorem 2.2.10. Let be a Z-graded unitary magma and θ be a rank function of . The map F θ is an operad morphism from C to RatFct.

Proof. For the sake of brevity of notation, for all positive integers < , we denote by

U the sums + • • • + -1 .
Let p and q be two -cliques of respective arities and , and ∈ [ ] be an integer. From the definition of the partial composition of C , the one (see (2.2.29)) of RatFct, and the fact that θ is a unitary magma morphism, we have

F θ (p) • F θ (q) = (F θ (p)) ( 1 -1 U + + + -1 ) (F θ (q)) ( + -1 ) =   1 < -1 U θ(p( ))     +1 < +1 U θ(p( )) + -1 + -1   U θ(p ) +   1 < +1 U θ(q( )) + -1 + -1   =   1 < -1 U θ(p( ))     +1 < +1 U θ(p( )) + -1 + -1   U θ(p )+θ(q 0 ) +     1 < +1 ( ) =(1 +1) U θ(q( )) + -1 + -1     =   1 < -1 U θ(p( ))     +1 < +1 U θ(p( )) + -1 + -1   U θ(p q 0 ) +     1 < +1 ( ) =(1 +1) U θ(q( )) + -1 + -1     = ( )∈ p• q U θ((p• q)( )) = F θ (p • q) (2.2.33)
Moreover, since θ(1 ) = 0, one has F θ ( ) = 1, so that F θ sends the unit of C to the unit of RatFct. Therefore, F θ is an operad morphism.

The operad morphism F θ is not injective. Indeed, by considering the magma Z together with its identity map Id as rank function, one has for instance . Now, since a Laurent polynomial is a linear combination of some Laurent monomials, by the linearity of F Id , the statement of the proposition follows.

F Id 1 -1 - 1 = ( 1 + 2 ) - 1 - 2 = 0 (2.2.34a) F Id -1 -1 --1 -1 - -1 -1 = 1 2 3 - 1 ( 2 + 3 ) 3 - 1 2 ( 2 + 3 ) = 0 (2.2.
For any 1, let

: C ( ) ⊗ C ( ) → C ( ) (2.2.36)
be the product defined for all -cliques p and q by (p q)(

) := p( ) q( ) (2.2.37)
where ( ) is any arc such that 1 < + 1, and then extended linearly. For instance, in CZ,

2 -1 1 -2 3 1 -1 1 2 = 3 3 -1 2 -1
(2.2.38) Proposition 2.2.12. Let be a Z-graded unitary magma and θ be a rank function of . For any homogeneous elements and of C of the same arity,

F θ ( )F θ ( ) = F θ ( ) (2.2.39)
Proof. Let p and q be two -cliques of C of arity . By definition of the operation on C ( ) and the fact that θ is a unitary magma morphism,

F θ (p)F θ (q) =   ( )∈ p + • • • + -1 θ(p( ))     ( )∈ q + • • • + -1 θ(q( ))   = 1 < +1 + • • • + -1 θ(p( ))+θ(q( )) = 1 < +1 + • • • + -1 θ(p( ) * q( )) = F θ (p q) (2.2.40)
By the linearity of F θ and of , (2.2.39) follows.

Proposition 2.2.13. Let p be an -clique of CZ. Then,

1 F Id (p) = F Id ((Cη)(p)) (2.2.41)
where η : Z → Z is the unitary magma morphism defined by η( ) := -for all ∈ Z.

Proof. Observe that (Cη)(p) is the -clique obtained by relabeling each arc ( ) of p by -p(

). Hence, since η is a unitary magma morphism, we have

F Id ((Cη)(p)) = ( )∈ p + • • • + -1 θ(-p( )) = ( )∈ p + • • • + -1 -θ(p( )) = 1 F Id (p)
(2.2.42) as expected.

Q

We define here quotients and suboperads of C , leading to the construction of some new operads involving various combinatorial objects which are, basically, -cliques with some restrictions.

Main substructures. Most of the natural subfamilies of

-cliques that can be described by simple combinatorial properties as -cliques with restrained labels for the bases, edges, and diagonals, white -cliques, -cliques with a fixed maximal crossing number, -bubbles, -cliques with a fixed maximal value for their degrees, nesting-free -cliques, and acyclic -cliques inherit from the algebraic structure of operad of C and form quotients and suboperads of C (see Table 1). We construct and briefly study T . Operads constructed as suboperads or quotients of C . All these operads depend on a unitary magma which has, in some cases, to satisfy some precise conditions. Some of these operads depend also on a nonnegative integer or subsets B, E, and D of . here these main substructures of C . and a partial composition r := p • q for a valid integer . By the definition of the partial composition of C , the base of r has the same label as the base of p, and all edges of r have labels coming from the ones of p and q. Moreover, all diagonals of r are either non-solid, or come from diagonals of p and q, or are the diagonal r( + |q|) which is labeled by p q 0 . Since 1 ∈ D, p ∈ E, q 0 ∈ B, and E B ⊆ D, all the labels of these diagonals are in D. For these reasons, r is in Lab B E D . This implies the statement of the proposition. is the linear span of all -cliques p such that cros(p) .

Proposition 3.1.3. Let be a unitary magma and 0 be an integer. Then, the space Cro is a quotient operad of C and is isomorphic to the suboperad of C restricted to the subspace generated by all -cliques with crossing numbers no greater than .

Proof. We first prove that Cro is a quotient of C . For this, observe that if p and q are two -cliques, cros(p • q) = max{cros(p) cros(q)} (3. To prove the second part of the statement, consider two -cliques p and q of Cro . By (3.1.5), all -cliques p • q are still in Cro , for all valid integers . Moreover, the unit of C belongs to Cro . This implies the desired property.

For instance, in the operad Cro 2 Z, we have Observe that Cro 0 is the linear span of all noncrossing -cliques. We can see these objects as noncrossing configurations [START_REF] Flajolet | Analytic combinatorics of non-crossing configurations[END_REF] where the edges and bases are colored by elements of and the diagonals by elements of ¯ . The operad Cro 0 has a lot of combinatorial and algebraic properties and will be studied in detail in [Gir18].

3.1.4. Bubbles. Let R Bub be the subspace of C generated by all -cliques that are not bubbles. As a quotient of graded vector spaces,

Bub := C /R Bub (3.1.7)
is the linear span of all -bubbles. Proposition 3.1.4. Let be a unitary magma. Then, the space Bub is a quotient operad of C . Proof. If p and q are two -cliques, all solid diagonals of p and q appear in p • q, for any valid integer . For this reason, if p is an -clique of R Bub , each -clique obtained by a partial composition involving p and other -cliques is still in R Bub . This proves that R Bub is an operad ideal of C and implies the statement of the proposition.

For instance, in the operad BubZ, we have 1 2 is the linear span of all -cliques p such that degr(p) .

• 2 1 = 1 1 2 (3.1.8a) -1 2 • 3 1 1 = 1 2 (3.1.8b) 1 2 • 3 1 = 0 (3.1.8c) 1 2 • 2 1 2 = 0 (3.1.8d) When is
Proposition 3.1.5. Let be a unitary magma without nontrivial unit divisors and 0 be an integer. Then, the space Deg is a quotient operad of C .

Proof. Since has no nontrivial unit divisors, for any -cliques p and q of C , each solid arc of p (resp. q) gives rise to a solid arc in p • q, for any valid integer . Hence, degr(p • q) max{degr(p) degr(q)} (3. Observe that Deg 0 is the linear span of all -cliques without solid arcs. If p and q are such -cliques, all partial compositions p • q are equal to the unique -clique without solid arcs of arity |p| + |q| -1. For this reason, Deg 0 is the associative operad As.

Any skeleton of an

-clique of arity of Deg 1 can be seen as a partition of the set [ + 1] into singletons or pairs. Therefore, Deg 1 can be seen as an operad on such colored partitions, where each pair of the partitions has one color from the set ¯ . In the operad Deg 1 D 0 (observe that D 0 is the only unitary magma without nontrivial unit divisors on two elements), one has for instance is the linear span of all nesting-free -cliques.

Proposition 3.1.6. Let be a unitary magma without nontrivial unit divisors. Then, the space Nes is a quotient operad of C .

Proof. Since has no nontrivial unit divisors, for any -cliques p and q of C , each solid arc of p (resp. q) gives rise to a solid arc in p • q, for any valid integer . For this reason, if p is an -clique of R Nes , p is not nesting-free and each -clique obtained by a partial composition involving p and other -cliques is still not nesting-free and thus, belongs to R Nes . This proves that R Nes is an operad ideal of C and implies the statement of the proposition. Lemma 3.1.7. Let be a finite unitary magma without nontrivial unit divisors. For all 2, the set of all -cliques of Nes ( ) is in one-to-one correspondence with the set of all Dyck paths of size + 1 wherein letters a at even positions are colored by ¯ . Moreover, there is a correspondence between these two sets that sends any -clique of Nes ( ) with solid edges to a Dyck path with exactly letters a at even positions, for any 0 .

Proof. In this proof, we denote by a the letter a of a Dyck path colored by ∈ ¯ . Given an -clique p of Nes ( ), we decorate each vertex of p by

(1) aa if has one outcoming arc and no incoming arc, where is the label of the outcoming arc from ;

(2) bb if has no outcoming arc and one incoming arc;

(3) ba if has both one outcoming arc and one incoming arc, where is the label of the outcoming arc from ; (4) ab otherwise.

Let φ be the map sending p to the word obtained by concatenating the decorations of the vertices of p thus described, read from 1 to + 1.

We show that φ is a bijection between the two sets of the statement of the lemma. First, observe that since p is nesting-free, for each vertex of p, there is at most one incoming arc to and one outcoming arc from . For this reason, for any vertex of p, the total number of incoming arcs to vertices of p is smaller than or equal to the total number of outcoming arcs to vertices of p, and the total number of vertices having an incoming arc is equal to the total number of vertices having an outcoming arc in p. Thus, by forgetting the colorings of its letters, the word φ(p) is a Dyck path.

Besides, given a Dyck path of size + 1 wherein letters a at even positions are colored by ¯ , one can build a unique -clique p of Nes ( ) such that φ(p) = . Indeed, by reading the letters of two by two, one knows the number of outcoming and incoming arcs for each vertex of p. Since p is nesting-free, there is one unique way to connect these vertices by solid diagonals without creating nestings of arcs. Moreover, by (1), (2), (3), and (4), the colors of the letters a at even positions allow to label the solid arcs of p. Hence φ is a bijection as claimed.

Finally, by definition of φ, we observe that if p has exactly solid arcs, the Dyck path φ(p) has exactly occurrences of the letter a at even positions. This implies the whole statement of the lemma.

Let nar(

) be the Narayana number [START_REF] Narayana | Sur les treillis formés par les partitions d'un entier et leurs applications à la théorie des probabilités[END_REF] defined for all 0 -2 by nar(

) := 1 + 1 -2 -1 (3.1.20)
The number of Dyck paths of size -1 and exactly occurrences of the factor ab is nar( ). Equivalently, this is also the number of binary trees with leaves and exactly internal nodes having an internal node as a left child. where := # . Proof. It is known from [Sul98] that the number of Dyck paths of size + 1 with occurrences of the letter a at even positions is the Narayana number nar( + 2 ). Hence, by using this property together with Lemma 3.1.7, we obtain that the number of nestingfree -cliques of size with solid arcs is ( -1) nar( + 2 ). Therefore, since a nesting-free -clique of arity can have at most solid arcs, (3.1.21) holds.

The skeletons of the -cliques of Nes of arities greater than 1 are the graphs such that, if { } and { } are two arcs such that < , then = and = . Therefore, Nes can be seen as an operad on such colored graphs, where the arcs of the graphs have one color from the set ¯ . Equivalently, as Lemma 3.1.7 shows, Nes can be seen as an operad of Dyck paths where letters a at even positions are colored by ¯ . Proposition 3.1.9. Let be a unitary magma without nontrivial unit divisors. Then, the space Acy is a quotient operad of C .

By

Proof. Since has no nontrivial unit divisors, for any -cliques p and q of C , each solid arc of p (resp. q) gives rise to a solid arc in p • q, for any valid integer . For this reason, if p is an -clique of R Acy , p is not acyclic and each -clique obtained by a partial composition involving p and other -cliques is still not acyclic and thus, belongs to R

Acy . This proves that R Acy is an operad ideal of C and implies the statement of the proposition. 3.2. Secondary substructures. Some more substructures of C are constructed and briefly studied here. They are constructed by mixing some of the constructions of the seven main substructures of C defined in Section 3.1 in the following sense.

For any operad and operad ideals R 1 and R 2 of , the space R 1 + R 2 is still an operad ideal of , and / (R 1 + R 2 ) is a quotient of both /R 1 and /R 2 . Moreover, if is a suboperad of and R is an operad ideal of , the space R ∩ is an operad ideal of , and / (R ∩ ) is a quotient of and a suboperad of /R. For these reasons (straightforwardly provable), we can combine the constructions of the previous section to build plenty new suboperads and quotients of C (see

Relations between the secondary and main substructures.

Here we list and explain the morphisms between the secondary and main substructures of C . First, immediately from their definitions, WNC is a suboperad of Cro

C

The clique construction provides alternative definitions of known operads. We explore here the cases of the operads MT and DMT of multi-tildes and double multi-tildes, and the gravity operad Grav. 4.1. Operads from language theory. We provide constructions of two operads coming from formal language theory by using the clique construction. DMT is an isomorphism between these two operads. Proof. There are two ways to prove the first assertion of the statement of the proposition.

On the one hand, this property follows from Proposition 2.1.2 and Proposition 4.1.1. On the other hand, the whole statement of the proposition is a direct consequence of the definition (4.1.10) of φ DMT , showing that φ DMT is an isomorphism of vector spaces, and, from the definitions of the partial compositions of DMT and CD 2 0 showing that φ DMT is an operad morphism. By Proposition 4.1.2, one can interpret the partial composition (4.1.9) of double multitildes as a partial composition of D 2 0 -cliques. This gives A gravity chord diagram is a { }-configuration c, where is any symbol, satisfying the following conditions. By denoting by the size of c, all the edges and the base of c are labeled (by ), and if ( ) and ( ) are two labeled crossing diagonals of c such that < , the arc ( ) is not labeled. In other words, the quadrilateral formed by the vertices , , , and of c is such that its side ( ) is unlabeled. For instance, (4.2.1) is a gravity chord diagram of arity 7 having four labeled diagonals (observe in particular that, as required, the arc (3 5) is not labeled). For any 2, Grav( ) is the linear span of all gravity chord diagrams of size . Moreover, Grav(1) is the linear span of the singleton containing the only polygon of size 1 where its only arc is not labeled. The partial composition of Grav is defined graphically as follows. For any gravity chord diagrams c and d of respective arities and , and ∈ [ ], the gravity chord diagram c • d is obtained by gluing the base of d onto the th edge of c, so that the arc ( + ) of c • d is labeled. For example,

• 3 = (4.2.2)
Among its most notable ones, C admits several quotients involving combinatorial families of decorated cliques, and contains some already studied operads. Let us address here some open questions.

First, we have for the time being no formula to enumerate prime (resp. white prime, minimal prime) -cliques (see (2.2.6) (resp. (2.2.7), (2.2.8)) for # = 2). Obtaining these forms a first combinatorial question.

When

is a Z-graded unitary magma, a link between C and the operad of rational functions RatFct has been developed in Section 2.2.8 by means of a morphism F θ between these two operads. We have observed that F θ is not injective (see (2.2.34a) and (2.2.34b)). A description of the kernel of F θ , even when is the unitary magma Z, seems not easy to obtain. Trying to obtain this description is a second perspective of this work.

Here is a third perspective. In Section 3, we have defined and briefly studied some quotients and suboperads of C . In particular, we have considered the quotient Deg 1 of C , involving -cliques of degree at most 1. As mentioned, Deg 1 D 0 is an operad defined on the linear span of involutions (except the nontrivial involution of S 2 ). A complete study of this operad seems worth considering, including a description of a minimal generating set, a presentation by generators and relations, a description of its partial composition on the H-basis and on the K-basis, and a realization of this operad in terms of standard Young tableaux.

I

  April 26, 2021. 2010 Mathematics Subject Classification. 05C76, 18D50, 05E99. Key words and phrases. Configuration of chords; Graph; Operad. 1 Configurations of chords on regular polygons are very classical combinatorial objects.

  General notations and conventions. All the algebraic structures of this article have a field of characteristic zero K as ground field. For any set S, K S denotes the linear span of the elements of S. For any integers and , [ ] denotes the set { ∈ N : } and [ ], the set [1 ]. The cardinality of a finite set S is denoted by #S. If is a word, its letters are indexed from left to right from 1 to its length | |. If is a letter, | | denotes the number of occurrences of in .

2.2. 1 .

 1 Binary relations. Let us start by remarking that, depending on the cardinality of , the set of all -cliques can be interpreted as particular binary relations. When 4, let us set = {1 a b c } so that a, b, and c are distinguished pairwise distinct elements of different from 1 . Given an -clique p of arity 2, we build a binary relation R on [ + 1] satisfying, for all

3.1. 1 .

 1 Restricting the labels. In what follows, if X and Y are two subsets of , X Y denotes the set { : ∈ X and ∈ Y }. Let B, E, and D be three subsets of and Lab B E D be the subspace of C generated by all -cliques p such that the bases of p are labeled by B, all edges of p are labeled by E, and all diagonals of p are labeled by D. Proposition 3.1.1. Let be a unitary magma and B, E, and D be three subsets of . If 1 ∈ B, 1 ∈ D, and E B ⊆ D, Lab B E D is a suboperad of C . Proof. First, since 1 ∈ B, the unit of C belongs to Lab B E D . Consider now two -cliques p and q of Lab B E D

  Let be a unitary magma and B, E, and D be three finite subsets of . where := #B, := #E, and := #D. Proof. By Proposition 2.2.1, there are ( +1 2 ) -cliques of arity , where := # . Hence, there are ( +1 2 ) / +1 -cliques of arity with all edges and the base labeled by 1 . This also says that there are ( +1 2 ) / +1 -cliques of arity with all diagonals labeled by D and all edges and the base labeled by 1 . Since an -clique of Lab B E D ( ) has its edges labeled by E and its base labeled by B, (3.1.1) follows. 3.1.2. White cliques. Let Whi be the subspace of C generated by all white -cliques. Since, by definition of white -cliques, Whi = Lab {1 } {1 } (3.1.2) by Proposition 3.1.1, Whi is a suboperad of C . It follows from Proposition 3.1.2 that when is finite, the dimensions of Whi satisfyRestricting the crossings. Let 0 be an integer and R Cro be the subspace of C generated by all -cliques p such that cros(p) + 1. As a quotient of graded vector spaces, Cro := C /R Cro (3.1.4)

  and suboperads of C . First, since any -clique of Cro is also an -

  .5. Restricting the degrees. Let 0 be an integer and R Deg be the subspace of C generated by all -cliques p such that degr(p) + 1. As a quotient of graded vector spaces

  1.11) and then, if p is an -clique of R Deg , each -clique obtained by a partial composition involving p and other -cliques is still in R Deg . This proves that R Deg is an operad ideal of C and implies the statement of the proposition. For instance, in the operad Deg 3 D 2 (observe that D 2 is a unitary magma without nontrivial unit divisors)

  solid arc ( ) of an -clique p of Deg 1 D 0 of arity as the transposition exchanging the letter and the letter , we can interpret p as an involution of S +1 made of the product of these transpositions. Hence, Deg 1 D 0 can be seen as an operad on involutions. Under this point of view, the partial compositions (3.1.13a) and (3.1.13b) translate on the Robinson-Schensted correspondence (see for instance [Lot02]), Deg 1 D 0 is an operad on standard Young tableaux. The dimensions of the operad Deg 1 except for the first terms, Sequence A000085 of [Slo]. Moreover, when # = 3form, except for the first terms, Sequence A047974 of [Slo]. Besides, any skeleton of an -clique of Deg 2 can be seen as a thunderstorm graph, i.e., a graph where connected components are cycles or paths. Therefore, Deg 2 can be seen as an operad on such colored graphs, where the arcs of the graphs have one color from the set ¯ . When # = 2, the dimensions of this operad begin by 1 8 41 253 1858 15796 152219 1638323 (3.1.17) and form, except for the first terms, Sequence A136281 of [Slo]. 3.1.6. Nesting-free cliques. Let R Nes be the subspace of C generated by all -cliques that are not nesting-free. As a quotient of graded vector spaces, Nes := C /R Nes (3.1.18)

For

  that in the same way as considering -cliques of crossing numbers or less leads to quotients Cro of C (see Section 3.1.3), it is possible to define analogous quotients Nes spanned by -cliques having solid arcs that nest at most other ones. Recall that a Dyck path of size is a word on {a b} of length 2 such that | | a = | | b and, for each prefix of , | | a | | b .

  Proposition 3.1.8, when # = 2, the dimensions of Nes begin by 1 5 14 42 132 429 1430 4862 (3.1.22) and form, except for the first terms, Sequence A000108 of [Slo]form, except for the first terms, Sequence A001003 of [Slo]. When # = 4, the dimensions of Nes begin by 1 19 100 562 3304 20071 124996 793774 (3.1.24) and form, except for the first terms, Sequence A007564 of [Slo]. 3.1.7. Acyclic decorated cliques. Let R Acy be the subspace of C generated by all cliques that are not acyclic. As a quotient of graded vector spaces, Acy := C /R Acy (3.1.25) is the linear span of all acyclic -cliques.

For

  the -cliques of Acy of arities greater than 1 are acyclic graphs or equivalently, forests of non-rooted trees. Therefore, Acy can be seen as an operad on colored forests of trees, where the edges of the trees of the forests have one color from the set ¯ . When # = 2, the dimensions of Acy begin by 1 7 38 291 2932 36961 561948 10026505 (3.1.27) and form, except for the first terms, Sequence A001858 of [Slo].

  is a suboperad of Mot and a quotient of WNC , and Luc is both a quotient of Bub and Deg 1 to Nes , from For to Mot , and from Mot to Luc . The relations between the secondary suboperads and quotients of C built here are summarized in the diagram of operad morphisms of Figure3of the secondary suboperads and quotients of C together with some of their related main suboperads and quotients of C . Arrows (resp. ) are injective (resp. surjective) operad morphisms. Here, is a unitary magma without nontrival unit divisors.

By<φ(

  Proposition 4.1.1, one can interpret the partial compositions (4.1.4a) and (4.1.4b) of multi-tildes as partial compositions of D 0 -cliques. This give respectively .2. Double multi-tildes. Double multi-tildes are natural generalizations of multi-tildes, introduced in [GLMN16]. A double multi-tilde is a triple ( s t) where ( t) and ( s) are both multi-tildes of the same arity . The arity of the double multi-tilde ( s t) is . As shown in [GLMN16], the linear span of all double multi-tildes admits a structure of an operad. This operad, denoted by DMT, is defined as follows. For any 1, DMT( ) is the linear span of all double multi-tildes of arity and the partial composition ( s t)• ( u v), ∈ [ ], of two double multi-tildes ( s t) and ( u v) is defined linearly by( s t) • ( u v) := ( s • u t • v) (4.1.8)where the two partial compositions • of the right member of (4.1.8) are the ones of MT. We can observe that DMT is isomorphic to the Hadamard product MT * MT. For instance, one has(3 {(2 2)} {(1 2) (1 3)}) • 2 (2 {(1 1)} {(1 2)}) = (4 {(2 2) (2 3)} {(1 3) (1 4) (2 3)}) (4.1.9) The unit of DMT is (1 ∅ ∅).Consider now the operad CD 2 0 and let φ DMT : DMT → CD 2 0 be the linear map defined as follows. The image by φ DMT of (1 ∅ ∅) is the unit of CD 2 0 and, for any double multi-tilde ( s t) of arity 2, φ DMT (( s t)) is the D 2 0 -clique of arity defined, for any 1 DMT ((4 {(2 2) (2 3)} {(1 3) (1 4) (2 3)})) = The operad CD 2 0 is isomorphic to the suboperad of DMT consisting in the linear span of all double multi-tildes except the three nontrivial double multi-tildes of arity 1. Moreover, φ

.

  Gravity operad. The operad of gravity chord diagrams Grav is an operad defined in[START_REF] Alm | Brown's dihedral moduli space and freedom of the gravity operad[END_REF]. This operad is the nonsymmetric version (obtained by forgetting the actions of the symmetric groups) of the gravity operad, a symmetric operad introduced by Getzler[START_REF] Getzler | Two-dimensional topological gravity and equivariant cohomology[END_REF]. Let us describe this operad.

  In this case, we say that is a set-operad with respect to the basis B.

	Moreover, when all the maps
	• : B( ) → B( + -1)	1 ∈ [ ] ∈ B( )	(1.1.5)
	defined by
	• ( ) = •	∈ B( )	(1.1.6)
	are injective, we say that B is a basic set-operad basis of . This notion is a slightly
	modified version of the original notion of basic set-operads introduced by Vallette [Val07].
	Finally, is cyclic (see [GK95]) if there is a map
	ρ : ( ) → ( )	1	(1.1.7)

). A symmetry of is either an automorphism or an antiautomorphism of . The set of all symmetries of forms a group for the map composition, called the group of symmetries of . A basis B := 1 B( ) of is a set-operad basis if all partial compositions of elements of B belong to B.

  If p has no labeled diagonals, p is a bubble. A triangle is a configuration of size 2. Obviously, all triangles are bubbles, and all bubbles are noncrossing.

		Two (non-necessarily labeled) diagonals (	) and (	) of p are crossing if
	<	< <	or	< < < . The crossing number of a labeled diagonal (	) of
	p is the number of labeled diagonals (	) such that (	) and (	) are crossing. The
	crossing number cros(p) of p is the maximal crossing among its labeled diagonals. When
	cros(p) = 0, there are no crossing diagonals in p and in this case, p is noncrossing. A
	(non-necessarily labeled) arc (	) is nested in a (non-necessarily labeled) arc (	) of p
	if	<	. We say that p is nesting-free if for any labeled arcs (	) and (	) of p
	such that (	) is nested in (	), (	) = (	). Besides, p is acyclic if skel(p) is acyclic,
	that is there is no subset {	1	} of [ + 1] of cardinality	3 such that {	+1 } and
	{	1 } are arcs in skel(p) for all ∈ [ -1]. When p has no labeled edges nor labeled
	base, p is white.

  generating set.Except for the first terms, the second one forms Sequence A006125, the third one forms Sequence A047656, and the last one forms Sequence A053763 of [Slo].

	Lemma 2.2.2. Let	be a unitary magma, p be an	-clique of arity	2, and (	)
	be a diagonal of p. Then, the following two assertions are equivalent:
	Proposition 2.2.1. Let	be a finite unitary magma. For all	2,
				dim C ( ) = ( +1 2 )	(2.2.2)
	where	:= # .
	Proof. By definition of the clique construction and of -cliques, the dimension of C ( )
	is the number of maps from the set (	) ∈ [ + 1] 2	: <	to	. Therefore, when
	2, this implies (2.2.2).
	From Proposition 2.2.1, the first dimensions of C	depending on	:= #	are
				1 1 1 1 1 1 1 1	= 1	(2.2.3a)
		1 8 64 1024 32768 2097152 268435456 68719476736	= 2	(2.2.3b)
	1 27 729 59049 14348907 10460353203 22876792454961
					150094635296999121	= 3 (2.2.3c)
	1 64 4096 1048576 1073741824 4398046511104 72057594037927936
					4722366482869645213696	= 4 (2.2.3d)

  of the statement of Lemma 2.2.2. Therefore, by this lemma, there are two -cliques q and r and an integer ∈ [|p|] such that |q| < |p|, |r| < |p|, and p = q • r.

	Computer experiments tell us that, when	:= #	= 2, the first numbers of prime
	-cliques are, size by size,
				0 8 16 352 16448 1380224	(2.2.6)
	Moreover, remark that the th term of this sequence is divisible by	+1	since the labels
	of the base and the edges of an	-clique p have no influence on the fact that p is prime.
	This gives the sequence
					0 1 1 11 257 10783	(2.2.7)
	enumerating the first of them size by size. Besides, a prime	-clique p is minimal if
	any	-clique obtained from p by replacing a solid arc by a non-solid one is not prime.
	Of course, all minimal prime	-cliques are white. Computer experiments show us that
	when	:= # = 2, the numbers of minimal prime	-cliques begin by
					0 1 1 5 22 119	(2.2.8)
	None of these sequences appear in [Slo] at this time.
	2.2.3. Associative elements.
	Proposition 2.2.4. Let	be a unitary magma and be an element of C (2) of the
	form
					:=	λ p p	(2.2.9)
					p∈
	Proposition 2.2.3. Let where the λ	be a unitary magma. The set	is a minimal generating
	set of C .
	Proof. We show by induction on the arity that	is a generating set of C . Let p be an
	-clique. If p is of arity 1, p =	and hence p trivially belongs to (C )	(recall that
	this notation stands for the suboperad of C	generated by	). Let us assume that p
	is of arity	2. First, if p ∈	, then p ∈ (C ) . Otherwise, p is an	-clique which
	satisfies the description By induction hypothesis, q and r belong to (C )	and hence, p also belongs to (C ) .

Finally, by Lemma 2.2.2, if p is a prime -clique, p cannot be expressed as a partial composition of prime -cliques. Moreover, since the space C (1) is trivial, these arguments imply that is a minimal generating set of C . p , p ∈ , are coefficients of K. Then, is associative if and only if

  When θ is a unitary magma automorphism of , since by Theorem 2.1.1 C is a functor respecting bijections, Cθ is an operad automorphism of C . Hence, Cθ belongs to the group of symmetries of C . Moreover, the fact that ref belongs to the group of symmetries of C can be established by showing that this map is an antiautomorphism of C , directly from the definition of the partial composition of C and that of ref.

	Graphically, ref(p) is the -clique obtained by applying on p a reflection trough the vertical
	line passing by its base. For instance, one has in CZ,
							
					-2	-2
				ref	 	-2 1	  =	1 -2	(2.2.15)
					1		1
	Proposition 2.2.5. Let	be a unitary magma. Then, the group of symmetries of C
	contains the map ref and all the maps Cθ where θ is a unitary magma automorphisms
	of	.
	Proof. 2.2.5. Basic set-operad basis. A unitary magma	is right cancelable if for any	∈
	,	=	implies = .
	Proposition 2.2.6. Let	be a unitary magma. The fundamental basis of C	is a
	basic set-operad basis if and only if	is right cancelable.
							13b)
	of CD 0 are associative.
	2.2.4. Symmetries. Let ref : C	→ C	be the linear map sending any	-clique p of
	arity to the -clique ref(p) of the same arity such that, for any arc (	) where 1	<
		+ 1,
				(ref(p)) (	) := p( -+ 2 -+ 2)	(2.2.14)

  Rational functions. The graded vector space of all commutative rational functions K(U), where U is the infinite commutative alphabet {

	2.2.8. 1	2	}, has the structure of an
	operad RatFct introduced by Loday [Lod10] and is defined as follows. Let RatFct( ) be the
	subspace K(	1	) of K(U) and
							RatFct :=	RatFct( )	(2.2.28)
							1
	Observe that since RatFct is a graded space, each rational function has an arity. Hence,
	by setting	1 (	1 ) := 1 and	2 (	1	2
							27b)

  1.5)for any valid integer . For this reason, if p is an -clique of R

					Cro	, each clique obtained
	by a partial composition involving p and other	-cliques is still in R Cro	. This proves
	that R Cro	is an operad ideal of C	and hence, that Cro	is a quotient of C .

  Table 2). Relations between substructures. The suboperads and quotients of C constructed in Sections 3.1 and 3.2 are linked by injective or surjective operad morphisms. To establish these, we begin with the following lemma. Acy , by definition, p has a cycle formed by solid arcs. Hence, p has in particular a solid arc and a vertex of degree 2 or more. For this reason, since R Whi is a suboperad of Lab B E D . Finally, there is a surjective operad morphism from Whi to the associative operad As sending any -clique p of Whi to the unique basis element of As of the same arity as the one of p. The relations between the main suboperads and quotients of C built here are summarized in the diagram of operad morphisms of Figure2.

	Operad 3.2.3. Colored forests. When 3.2.6. Colored Lucas configurations. When Objects is a unitary magma without nontrivial unit divisors, let is a unitary magma without nontrivial unit 2 , . . . , , 3, be the vertices of p that form this cycle. We can assume without loss Ideal of C WNC White noncrossing cliques R Cro 0 ∩ Whi Pat Forests of paths R Deg 2 + R Acy For Forests R Cro 0 + R Acy Mot Motzkin configurations R Cro 0 + R Deg 1 Dis Dissections of polygons R Cro 0 + R Deg 1 For := C / R Cro 0 divisors, let of generality that 1 for all ∈ [ ] and thus, that ( 1 2 ) and ( 1 ) are solid arcs of + R Acy (3.2.6) The skeletons of the -cliques of For are forests of rooted trees having no arcs { } and { } satisfying < < < . Therefore, For can be seen as an operad on such colored forests, where the edges of the forests have one color from the set ¯ Luc := C / R Bub + R Deg (3.2.12) p being part of the cycle. Then, when 2 < , since 1 1 < 2 , the arc ( 1 2 ) 1 The skeletons of the -cliques of Luc is nested in ( 1 ). Otherwise, < 2 , and since 1 1 < 2 , the arc ( 1 ) is are graphs such that all vertices are of degree at most 1 and all arcs are of the form { nested in ( 1 2 ). For these reasons, since R Nes is the linear span of all -cliques that + 1} or {1 + 1}, where + 1 is the number of vertices of the graphs. Therefore, Luc are non nesting-free, p is in R Nes . This implies (iv). can be seen as an operad on such colored . When # = 2, the dimensions of For begin by graphs, where the arcs of the graphs have one color from the set ¯ . When # = 2, the 3.3.1. Relations between the main substructures. Here we list and explain the mor-∩ Whi dimensions of Luc begin by phisms between the main substructures of C . First, Lemma 3.3.1 implies that there
	Luc are surjective operad morphisms from Acy Lucas configurations 1 7 33 81 1083 6854 45111 305629 R Bub 1 4 7 11 18 29 47 76 to Deg 1 , from Nes + R Deg 1	to Deg 0	(3.2.7) (3.2.13) , from
	and form, except for the first terms, Sequence A054727, of [Slo]. Bub to Deg 0 , from Cro 0 to Bub , from Deg 2 to Bub , from Deg 2 and form, except for the first terms, Sequence A000032 of [Slo]. and from Acy to Nes . Second, when B, E, and D are subsets of such that 1 ∈ B, to Nes ,
	T operads depend on a unitary magma . Operads obtained as quotients of C by mixing certain ideals of C . All these which has, in some cases, to satisfy some precise conditions. 3.2.1. Colored white noncrossing configurations. When is a unitary magma, let WNC := Whi /R Cro 0 ∩ Whi (3.2.1) 3.2.4. Colored Motzkin configurations. When is a unitary magma without nontrivial unit divisors, let Mot := C / R Cro 0 + R Deg 1 (3.2.8) 3.3. Lemma 3.3.1. Let be a unitary magma. Then, (i) the space R Deg 1 ; 1 ∈ E, and E B ⊆ D, C
	The The skeletons of the -cliques of WNC (ii) the spaces R When # = 2, the dimensions of WNC are white noncrossing -cliques of Mot are configurations of non-intersecting chords ; Deg 0 -cliques. begin by 1 1 3 11 45 197 903 4279 (3.2.2) on a circle. Equivalently, these objects are graphs of involutions (see Section 3.1.5) having no arcs { } and { } satisfying < (iii) the spaces R Cro 0 and R Deg 2 are subspaces of R Bub ; < < . These objects are enumerated by Motzkin numbers [Mot48]. Therefore, Mot graphs, where the arcs of the graphs have one color from the set ¯ . When # = 2, the Proof. All the spaces appearing in the statement of the lemma are subspaces of C can be seen as an operad on such colored (iv) the spaces R Deg 2 and R Acy are subspaces of R Nes . Acy Deg Cro Lab B E D
	dimensions of Mot generated by some subfamilies of begin by and form Sequence A001003 of [Slo]. When # -cliques. Therefore, to prove the assertions of the = 3, the dimensions of WNC begin lemma, we shall prove inclusions of adequate subfamilies of such objects. Deg 2 Cro 0
	by	If p is an	-clique of R	1 4 9 21 51 127 323 835		(3.2.9)
	1 1 5 31 215 1597 12425 99955 and form Sequence A269730 of [Slo]. Observe that these dimensions are shifted versions (3.2.3) and form, except for the first terms, Sequence A001006, of [Slo]. Deg 1 is the linear span of all -cliques of degree 2 or more, p is in R Deg 1 . This Nes Deg 1 Bub Whi
	implies (i). the ones of the γ-polytridendriform operads TDendr γ [Gir16b] with γ := # -1. 3.2.2. Colored forests of paths. When is a unitary magma without nontrivial unit divi-If p is an -clique of R Nes or R Bub , by definition, p has in particular a solid arc. 3.2.5. Colored dissections of polygons. When is a unitary magma without nontrivial unit divisors, let Hence, since R Deg 0 is the linear span of all -cliques with at least one vertex with a Deg 0 . This implies (ii). positive degree, p is in R Deg 0
	sors, let If p is an F The skeletons of the Dis := Whi / R Cro 0 -clique of R Cro 0 or R Deg . The diagram of the main suboperads and quotients of C . Arrows (resp. ) + R Deg 1 ∩ Whi , p has in particular a solid diagonal. Indeed, (3.2.10) 2 Pat := C / R Deg 2 + R Acy (3.2.4) -cliques of Pat are forests of non-rooted trees that are paths. graphs have one color from the set ¯ . is the number of vertices of the graphs. Therefore, Dis can be seen as an operad on . When span of all Bub . This implies (iii). -cliques with at least one solid diagonal, p is in R such colored graphs, where the arcs of the graphs have one color from the set ¯ (min{ } max{ Bub is the linear }) is a solid diagonal of p. For this reason, since R Therefore, Pat can be seen as an operad on such colored graphs, where the arcs of the The skeletons of the -cliques of Dis are strict dissections of polygons, that are graphs of Motzkin configurations with no arcs of the form { with = -1, = + 1, and = |p| + 1 for at least one ∈ [3], so that the arc + 1} or {1 + 1}, where + 1 when p is in R Cro 0 this property is immediate. When p is in R Deg 2 are injective (resp. surjective) operad morphisms. Here, is a unitary magma without , since p has a vertex of degree 3 or more, the skeleton of p has three arcs { 1 }, { 2 }, and { nontrivial unit divisors, is a positive integer, and B, E, and D are subsets of such that 3 } 1 ∈ B, 1 ∈ E, and E B ⊆ D.
	When # = 2, the dimensions of Pat # = 2, the dimensions of Dis begin by If p is an -clique of R Deg 2 or R Acy , p has in particular a solid arc nested in another begin by
	1 7 34 206 1486 12412 117692 1248004 1 1 3 6 13 29 65 148 one. Indeed, when p is in R Deg 2 , since p has a vertex of a degree 3 or more, the skeleton (3.2.5) (3.2.11) of p has three arcs { 1 }, { 2 }, and { 3 }. One can check that for all relative orders
	an form, except for the first terms, Sequence A011800 of [Slo]. and form, except for the first terms, Sequence A093128 of [Slo]. between the vertices , 1 , 2 , and
						1 ,

Acy is a subspace of R Nes and R Bub are subspaces of R 3 , one of these arcs is nested in another one, so that p is not nesting-free. When p is in R Acy , p contains a cycle formed by solid arcs. Let

4.1.1. Multi-tildes. Multi-tildes are operators introduced in [CCM11] in the context of formal language theory as a convenient way to express regular languages. Let, for any 1, P be the set

A multi-tilde is a pair ( s) where is a positive integer and s is a subset of P . The arity of the multi-tilde ( s) is .

As shown in [LMN13], the graded (by the arity) collection of all multi-tildes admits a very natural structure of an operad. This operad, denoted by MT, is defined as follows. The partial composition ( s) • ( t), ∈ [ ], of two multi-tildes ( s) and ( t) is defined by

For instance, one has Observe that the multi-tilde (1 ∅) is the unit of MT.

Let φ MT : MT → CD 0 be the linear map defined as follows. For any multi-tilde ( s)

The operad CD 0 is isomorphic to the suboperad of MT consisting in the linear span of all multi-tildes except the nontrivial multi-tilde (1 {(1 1)}) of arity 1.

Moreover, φ

MT is an isomorphism between these two operads. Proof. A direct consequence of the definition (4.1.5) of φ MT is that this map is an isomorphism of vector spaces. Moreover, it follows from the definitions of the partial compositions of MT and CD 0 that φ MT is an operad morphism. Grav is an isomorphism between these two operads.

Proof. Let us denote by

Grav the subspace of CD 0 described in the statement of the proposition. First of all, it follows from the definition of the partial composition of CD 0 that

Grav is closed under the partial composition operation (this property can be also seen as a consequence of the fact that the partial composition of two gravity chord diagrams is still a gravity chord diagram [START_REF] Alm | Brown's dihedral moduli space and freedom of the gravity operad[END_REF]). Hence, and since Grav contains the unit of CD 0 , Grav is an operad. Second, observe that the image of φ Grav is the underlying space of Grav and, from the definition of the partial composition of Grav, one can check that φ Grav is an operad morphism. Finally, since φ

Grav is a bijection from Grav to Grav , the statement of the proposition follows. Proposition 4.2.1 shows hence that the operad Grav can be built through the clique construction. Moreover, as explained in [START_REF] Alm | Brown's dihedral moduli space and freedom of the gravity operad[END_REF], Grav contains the nonsymmetric version of the Lie operad, the symmetric operad describing the category of Lie algebras. This nonsymmetric version of the Lie operad as been introduced in [ST09]. Since Lie is contained in Grav as the subspace of all gravity chord diagrams having the maximal number of labeled diagonals for each arity, Lie can be built through the clique construction as the suboperad of CD 0 containing all the D 0 -cliques that are images by φ Grav of such maximal gravity chord diagrams.

Besides, this alternative construction of Grav leads to the following generalization for any unitary magma of the gravity operad. Let Grav be the linear span of all -cliques satisfying the gravity condition. It follows from the definition of the partial composition of C that Grav is an operad. Moreover, observe that when has nontrivial unit divisors, Grav is not a free operad.

C

This work presents and studies the clique construction C, producing operads from unitary magmas. We have seen that C has many both algebraic and combinatorial properties.


Alain Largeteau

Stéphane Darracq

Graziella Goglio

Gérard Demazeau

Solvothermal crystal growth of functional materials

Keywords: solvothermal process, crystal growth, solvent, nutrient, thermodynamical parameters, chemical parameters, density of defects

Hydrothermal crystal growth was associated with the development of α-quartz single crystals for devices based on piezoelectric materials. This technology was then used to elaborate single crystals of different functional materials such as quartz-like oxides (AlPO 4 , GaPO 4 , GaAsO 4 , etc.), calcite CaCO 3 , etc. During the last few years two materials, GaN (using non-aqueous solvent) and ZnO, have been particularly investigated due to their specific physical properties appropriate to the development of new technologies. On the basis of the different chemical and thermodynamical parameters governing the solvothermal crystal growth (nature and purity of the nutrient, nature of the solvent, pressure, temperature and thermal gradient), a discussion will be presented on the trends relative to crystal growth technology, in particular, for growing oxide and non-oxide materials.

Introduction

Hydrothermal processing of materials has originated from geological research [START_REF] Yoshimura | Hydrothermal processing of materials: past, present, future[END_REF]. The industrial applications of α-quartz correlated to its piezoelectric properties have historically improved the development of hydrothermal crystal growth process [START_REF] Iwasaki | Historical review of quartz crystal growth[END_REF][START_REF] Laudise | Perfection of quartz and its connection to crystal growth[END_REF][START_REF] Demazeau | Hydrothermal crystal growth of α-quartz: new specificities correlated to applications[END_REF][START_REF] Buisson | Hydrothermal growth of quartz crystals in industry, present status and evolution[END_REF][START_REF] Hervey | Synthetic quartz -a review[END_REF][START_REF] Hickel | The hydrothermal growth of α-quartz and the nature of the physico-chemical defects detected in the resulting crystal[END_REF][START_REF] Evseeva | Growth of piezoelectric quartz crystals of large dimension on seeds parallel to the (10-11) face and lengthened along the y-axis[END_REF]. During the twentieth century, this process was developed for the elaboration of different single crystals as quartz-like oxides (AlPO 4 , GaPO 4 , GaAsO 4 , etc.) [START_REF] Zvereva | Hydrothermal growth of OH-free AlPO 4 and GaPO 4 crystals, the way of twin reducing[END_REF][START_REF] Motchany | Crystal growth of an α-quartz-like piezoelectric material, berlinite[END_REF][START_REF] Jacobs | Physico-chemical aspect of the hydrothermal growth of GaPO 4[END_REF][START_REF] Yot | Advances in crystal growth and characterizations of gallium orthophosphate, GaPO 4[END_REF][START_REF] Cambon | Growth and dielectric characterization of large single crystals of GaAsO 4 , a novel piezoelectric material[END_REF], calcite, etc. [START_REF] Kikuta | Hydrothermal growth and dissolution behaviour of calcite single crystal in nitrate solutions[END_REF][START_REF] Yanagisawa | Hydrothermal single crystal growth of calcite in ammonium acetate solution[END_REF][START_REF] Borodin | Growth and characteristics of calcite single crystals[END_REF]. During these last 10 years two main materials, GaN [START_REF] Denis | Gallium nitride bulk crystal growth processes: a review[END_REF][START_REF] Fukuda | Prospects for the ammonothermal growth of large GaN crystals[END_REF] and ZnO [START_REF] Ehrentraut | Solvothermal growth of ZnO[END_REF][START_REF] Demianets | Status of hydrothermal crystal growth of bulk ZnO: latest issues and advantages[END_REF], have attracted new developments for solvothermal crystal growth processes (the solvent being aqueous or non-aqueous solutions).

This paper is devoted to the analysis of the influence of different parameters (thermodynamical: pressure, growth temperature, temperature gradient, chemical nature of the solvent and of the nutrient, chemical interaction between solvent and crystal growth vessel, etc.) on the quality of the single crystal (density of chemical and physical defects). In addition, a perspective on the development of solvothermal crystal growth processes will be given. *Corresponding author. Email: demazeau@icmcb-bordeaux.cnrs.fr

Principle of solvothermal crystal growth processes

Solvothermal crystal growth processes involve the chemical transport in a solvent (the solvent being in sub-or supercritical conditions) of chemical species issued from the partial dissolution of a chemical precursor (the nutrient) and the deposition of such species on seeds. In order to improve the solubility of the nutrient and consequently the concentration of the chemical species in the liquid phase solvent, pressure (for which the maximum value is correlated to pressure) is involved either as autogeneous pressure (pressure value depending then on the percentage of filling of the crystal growth vessel and the temperature) or as imposed pressure at the beginning of the crystal growth experiment.

A gradient of temperature between the nutrient and the seeds ensures the transportation of chemical species into the liquid phase through the convection phenomenon.

The variation of the nutrient solubility versus temperature (direct or reverse solubility) is an important parameter in order to get a supersaturated solution close to the seeds; the dissolution temperature can be higher (direct) or smaller (reverse) than the crystallization temperature.

The principle of the solvothermal crystal growth process is schematically represented in Figure 1.

Role of the different parameters governing solvothermal crystal growth processes

These parameters can be divided into two sets:

-Thermodynamical parameters (pressure P , crystal growth temperature T growth , temperature gradient T ) -Chemical parameters (in particular the chemical nature of the nutrient and the solvent)

Role of the thermodynamical parameters

The objective of pressure is to control the solubility of the nutrient in mild temperature conditions (T < 500 • C). The temperature value, in particular, the growth temperature (T growth , which is T -T in the case of a direct solubility) appears, in some cases, to play an important role in the quality of the resulting single crystals.

Such a correlation has been underlined, in particular, for the hydrothermal crystal growth of α-quartz, the IR α 3500 coefficient being characteristic of the anionic substitution (O 2-→ OH -), which decreases when T growth is increased (Figure 2) [START_REF] Lafon | Evaluation du rôle des principaux paramètres gouvernant la cristallogénèse du quartz α sur la densité de défauts physiques et chimiques au sein de cristaux de synthèse[END_REF].

The T value controls -through the convection phenomenon -the transport of chemical species and consequently the kinetics of crystal growth if the solubility value is sufficient. The increase of T leading to the increase of the density of defects (chemical and physical defects), the temperature gradient is consequently optimized versus economical aspects (kinetics of crystal growth and the reduction of the required crystal growth duration) and the crystal quality (in particular the admitted density of defects depending on the industrial applications).

Role of the chemical factors

The required characteristics of the nutrient depend on the solvent and the pressure-temperature conditions in order to induce an appropriated value of the dissolution.

The nutrient can be natural (or synthetic) polycrystals with the same composition as that of the target single crystal. Nevertheless synthetic powders (preferably compacted) can also be used [START_REF] Largeteau | Solvothermal synthesis of α-quartz powder doped with germanium[END_REF].

The seeds can be wafers with the same composition and structure as the target single crystal (homoepitaxy) and in some cases (in absence of adequate wafers), seeds are selected among materials of same structure or close to the structure of the target single crystal but with a different composition (heteroepitaxy).

The solvent must be chemically compatible with the composition of the target single crystals, the use of different mineralizers being able to help the dissolution and deposition processes.

In addition the chemical composition of the solvent can be optimized in order to reduce or to vanish the corrosion phenomena with the internal walls of the reaction vessel.

Trends in solvothermal crystal growth

Historically, hydrothermal crystal growth processes were developed through the elaboration of α-quartz single crystals. In particular SPEZIA has brought an important contribution [START_REF] Spezia | La pressione e chimicamente inattive nella solubilite e riecostituzione del quarzo[END_REF] for developing hydrothermal crystal growth [START_REF] Laudise | Hydrothermal crystal growth[END_REF].

Hydrothermal crystal growth of α-quartz

During the past 50 years, the improvement of the quality of single crystals (low density of chemical and physical defects) has mobilized a lot of research works [START_REF] Hickel | The hydrothermal growth of α-quartz and the nature of the physico-chemical defects detected in the resulting crystal[END_REF]. Different solvents (NaOH, Na 2 CO 3 , NaHCO 3 , KOH, KCl, NaCl, NaF in aqueous solution, etc.) have been investigated [START_REF] Laudise | Hydrothermal crystal growth[END_REF].

The influence of pressure on the growth rate (mm/day) has been recently evaluated (Figure 3) -an improvement of the crystal growth rate being observed with increasing pressure.

Such a behavior seems to have no influence on the quality of the resulting α-quartz single crystals, due to the decreasing IR signal corresponding to hydroxyl groups into the α-quartz lattice (α 3500 ) versus the pressure value (Figure 4) [START_REF] Lafon | Evaluation du rôle des principaux paramètres gouvernant la cristallogénèse du quartz α sur la densité de défauts physiques et chimiques au sein de cristaux de synthèse[END_REF].

The influence of the crystal growth temperature has also been evaluated (Figure 2), the IR α 3500 coefficient characterizing roughly the anionic substitution (O 2-→ OH -), decreasing with the T growth value as previously mentioned and illustrated in Figure 2 [START_REF] Lafon | Evaluation du rôle des principaux paramètres gouvernant la cristallogénèse du quartz α sur la densité de défauts physiques et chimiques au sein de cristaux de synthèse[END_REF].

These characterizations suggest that new processes at high pressure and high temperature could improve the α-quartz qualities.

Quartz-like (III-V) O 4 materials such as AlPO 4 [START_REF] Zvereva | Hydrothermal growth of OH-free AlPO 4 and GaPO 4 crystals, the way of twin reducing[END_REF][START_REF] Motchany | Crystal growth of an α-quartz-like piezoelectric material, berlinite[END_REF], GaPO 4 [START_REF] Jacobs | Physico-chemical aspect of the hydrothermal growth of GaPO 4[END_REF][START_REF] Yot | Advances in crystal growth and characterizations of gallium orthophosphate, GaPO 4[END_REF], GaAsO 4 [START_REF] Cambon | Growth and dielectric characterization of large single crystals of GaAsO 4 , a novel piezoelectric material[END_REF], etc., have attracted attention in order to improve piezoelectric properties.

Solvothermal crystal growth of materials for optoelectronic applications

Two main materials have been investigated during the past few years: zinc oxide (ZnO) and gallium nitride (GaN) due to their physical properties (wide band-gap semiconductors, high exciton binding energy value) leading to potential applications in different domains (optoelectronic, spintronic, etc.).

Compared with α-quartz, the solvothermal crystal growth of these materials is characterized by the following features:

-the temperature: no limit is imposed by any structural transition, -the solvent: high concentrations of alkaline mineralizers in aqueous solution for ZnO and liquid ammonia for GaN are needed, inducing corrosion phenomena, -the nutrient and seeds: in both cases, natural single crystals do not exist. Because of differences in the nature of the chemical bonds in the nutrient (ionic for ZnO and covalent for GaN) and due to the nature of the involved solvent (aqueous solution and ammonia, respectively), the thermodynamical P , T conditions governing the solvothermal crystal growth are different between ZnO (300 < T < 400 • C, 30 < P < 100 MPa) and GaN (400 < T < 600 • C, 200 < P < 300 MPa).

Different review papers summarize the state-of-the-art solvothermal crystal growth of GaN [START_REF] Denis | Gallium nitride bulk crystal growth processes: a review[END_REF][START_REF] Fukuda | Prospects for the ammonothermal growth of large GaN crystals[END_REF] and ZnO [START_REF] Ehrentraut | Solvothermal growth of ZnO[END_REF][START_REF] Demianets | Status of hydrothermal crystal growth of bulk ZnO: latest issues and advantages[END_REF].

The main interests in the solvothermal crystal growth are: (i) the ability to be scaled up in order to improve the size of the wafers required for industrial developments [START_REF] Fukuda | Prospects for the ammonothermal growth of large GaN crystals[END_REF], (ii) to decrease of the density of defects (in particular the dislocation density) through the control of the thermodynamical parameters.

In order to reduce the corrosion phenomena, research into mild temperature conditions appears to be an important challenge, particularly in the case of future ZnO studies.

Consequently, two areas must be developed in the near future: (i) the use of nutrient characterized by a high chemical reactivity (in the case of gallium nitride, Li 3 GaN 2 was tested as

Figure 1 .

 1 Figure 1. Principle of the solvothermal crystal growth (the repartition of seeds and nutrient in the high pressure vessel corresponds to a direct solubility).

Figure 2 .

 2 Figure 2. Evaluation of the impact of the crystal growth temperature (T growth ) on the quality of α-quartz single crystals (solvent = NaOH 1 M, P = 150 MPa, T = 35 • C, duration = 10 days).

Figure 3 .

 3 Figure 3. Influence of pressure value on the growth rate of α-quartz (α 3500 ) (solvent: NaOH (1 M), crystal growth temperature: 350 • C, T = 10 • C).

Figure 4 .

 4 Figure 4. Evaluation through IR spectroscopy (α 3500 ) of the hydroxyl group concentration into the α-quartz lattice versus the pressure value selected for crystal growth.

Conclusions

Solvothermal crystal growth processes are important for developing single crystals of functional materials to be used in different industrial sectors. The main interests in such crystal growth techniques are: (i) the ability to improve the quality of the single crystals through the optimization of the chemical and thermodynamical parameters, (ii) the scale-up at the industrial level due to the large developments induced by the hydrothermal crystal growth of α-quartz.