Pierre Gillibert
email: gilliber@karlin.mff.cuni.cz

THE POSSIBLE VALUES OF CRITICAL POINTS BETWEEN STRONGLY CONGRUENCE-PROPER VARIETIES OF ALGEBRAS

Keywords: 2000 Mathematics Subject Classification. 06B10, 03C05 Variety of algebras, compact, congruence, critical point, condensate, lifter, Armature Lemma

We denote by Conc A the (∨, 0)-semilattice of all finitely generated congruences of an (universal) algebra A, and we define Conc V as the class of all isomorphic copies of all Conc A, for A ∈ V, for any variety V of algebras.

Let V and W be locally finite varieties of algebras such that for each finite algebra A ∈ V there are, up to isomorphism, only finitely many B ∈ W such that Conc A ∼ = Conc B, and every such B is finite. If Conc V ⊆ Conc W, then there exists a (∨, 0)-semilattice of cardinality ℵ 2 in (Conc V) -(Conc W). Our result extends to quasivarieties of first-order structures, with finitely many relation symbols, and relative congruence lattices.

In particular, if W is a finitely generated variety of algebras, then this occurs in case W omits the tame congruence theory types 1 and 5; which, in turn, occurs in case W satisfies a nontrivial congruence identity.

The bound ℵ 2 is sharp.

Introduction

Why do so many representation problems in algebra, enjoying positive solutions in the finite case, have counterexamples of minimal cardinality either ℵ 0 , ℵ 1 , or ℵ 2 , and no other cardinality? By a representation problem, we mean that we are given categories A and B together with a functor Φ : A → B, and we are trying to determine whether an object B of B is isomorphic to Φ(A) for some object A of A. We are also given a mapping from the objects of B to the cardinals, that behaves like the cardinality mapping on sets.

Examples of such representation problems cover various fields of mathematics. Here are a few examples, among many:

• Every (at most) countable Boolean algebra is generated by a chain (cf. [START_REF] Grätzer | Lattice Theory: Foundation[END_REF]Theorem 172]), but not every Boolean algebra is generated by a chain (cf. [START_REF] Grätzer | Lattice Theory: Foundation[END_REF]Lemma 179]). It is an easy exercise to verify that in fact, every subchain C of the free Boolean algebra F on ℵ 1 generators is countable, thus F cannot be generated by C. • Every dimension group with at most ℵ 1 elements is isomorphic to K 0 (R) for some (von Neumann) regular ring R (cf. [START_REF] Elliott | On the classification of inductive limits of sequences of semisimple finitedimensional algebras[END_REF][START_REF] Goodearl | Tensor products of dimension groups and K 0 of unitregular rings[END_REF]), but there is a dimension group with ℵ 2 elements which is not isomorphic to K 0 (R) for any regular ring R (cf. [START_REF] Wehrung | Non-measurability properties of interpolation vector spaces[END_REF]).

• Every distributive algebraic lattice with at most ℵ 1 compact elements is isomorphic to the congruence lattice of some lattice (cf. [START_REF] Huhn | On the representation of distributive algebraic lattices, I[END_REF][START_REF] Huhn | On the representation of distributive algebraic lattices[END_REF][START_REF] Huhn | On the representation of distributive algebraic lattices[END_REF]), but not every distributive algebraic lattice is isomorphic to the congruence lattice of some lattice (cf. [START_REF] Wehrung | A solution to Dilworth's congruence lattice problem[END_REF]); the minimal number of compact elements in a counterexample, namely ℵ 2 , is obtained in [START_REF] Růžička | Free trees and the optimal bound in Wehrung's theorem[END_REF].

In an earlier paper [START_REF] Gillibert | Critical points of pairs of varieties of algebras[END_REF], we introduced a particular case of the kind of representation problem considered above, concentrated in the notion of critical point between two varieties of (universal) algebras. It turned out that this notion often behaves as a paradigm for those kinds of problems. The present paper will be centered on that paradigm, and will offer an explanation, in that context, why for so many representation problems, the minimal size of a counterexample (if it exists at all) lies below ℵ 2 . Although initially stated for universal algebras, the method of proof of our main result (Theorem 5.1) carries a potential of generalization to many other contexts, starting with Theorem 6.1.

Let us be a bit more precise. For an algebra A we denote by Con c A the (∨, 0)semilattice of all compact (i.e., finitely generated) congruences of A. A lifting of a (∨, 0)-semilattice S is an algebra A such that Con c A ∼ = S. For a variety V of algebras we denote by Con c V the class of all (∨, 0)-semilattices with a lifting in V.

For varieties V and W of algebras, the critical point between V and W, denoted by crit(V; W), is the smallest cardinality of a member of (Con c V) -(Con c W) if Con c V ⊆ Con c W, and ∞ otherwise (cf. [START_REF] Gillibert | Critical points of pairs of varieties of algebras[END_REF][START_REF] Tůma | A survey of recent results on congruence lattices of lattices[END_REF]).

The critical point between two varieties can be anything we like. For example, for (possibly infinite) fields F and K with card F < card K, it is easy to verify that crit(F -vector spaces; K-vector spaces) = card F + 1 .

On the other hand, once we fix restrictions on the similarity types of our algebras, the situation becomes much more interesting. All known critical points between varieties of algebras, with a countable similarity type, are either ≤ ℵ 2 or equal to ∞. For example, it is proved in [START_REF] Růžička | Distributive congruence lattices of congruencepermutable algebras[END_REF] that crit(lattices; groups) = crit(lattices; rings) = ℵ 2 .

It is also easily seen that crit(groups; lattices) = 5. Ploščica in [START_REF] Ploščica | Non-representable distributive semilattices[END_REF], using methods introduced by Wehrung in [START_REF] Wehrung | A solution to Dilworth's congruence lattice problem[END_REF] and Růžička in [START_REF] Růžička | Free trees and the optimal bound in Wehrung's theorem[END_REF], finds a majority algebra M of cardinality ℵ 2 such that Con c M has no lifting by any lattice. However, every distributive (∨, 0)-semilattice of cardinality ≤ ℵ 1 is liftable by a lattice (cf. [START_REF] Huhn | On the representation of distributive algebraic lattices, I[END_REF][START_REF] Huhn | On the representation of distributive algebraic lattices[END_REF][START_REF] Huhn | On the representation of distributive algebraic lattices[END_REF]), therefore the critical point between the variety of all majority algebras and the variety of all lattices is ℵ 2 .

A strong restriction on the possible values of critical points between finitely generated congruence-distributive varieties of algebras is brought by the following dichotomy result proved in [START_REF] Gillibert | Critical points of pairs of varieties of algebras[END_REF].

Theorem 1.1. Let V be a locally finite variety of algebras and let W be a finitely generated congruence-distributive variety of algebras.

If Con c V ⊆ Con c W, then crit(V; W) < ℵ ω .
Critical points between varieties of lattices have been particularly studied. Tools for proving the countability of certain critical points are given, along with examples, in [START_REF] Ploščica | Separation in distributive congruence lattices[END_REF][START_REF] Ploščica | Relative separation in distributive congruence lattices[END_REF][START_REF] Gillibert | Critical points between varieties generated by subspace lattices of vector spaces[END_REF]. Examples of varieties of lattices with critical point ℵ 2 are given in [START_REF] Ploščica | Separation properties in congruence lattices of lattices[END_REF][START_REF] Ploščica | Dual spaces of some congruence lattices[END_REF][START_REF] Gillibert | Critical points between varieties generated by subspace lattices of vector spaces[END_REF]. In [START_REF] Gillibert | Critical points of pairs of varieties of algebras[END_REF] we give two finitely generated varieties of modular lattices with critical point ℵ 1 , solving a problem by Tůma and Wehrung in [START_REF] Tůma | A survey of recent results on congruence lattices of lattices[END_REF]. In [START_REF] Gillibert | The possible values of critical points between varieties of lattices[END_REF] we establish the following dichotomy result for varieties of lattices. Theorem 1.2. Let V and W be varieties of lattices such that every simple member of W contains a prime interval. If

Con c V ⊆ Con c W, then crit(V; W) ≤ ℵ 2 . Moreover, Con c V ⊆ Con c W if and only if V is contained in either W or its dual.
In particular, this solves the finitely generated case of (the correct recasting of) the critical point conjecture for lattices (cf. [START_REF] Tůma | A survey of recent results on congruence lattices of lattices[END_REF]Problem 5]). However the conjecture can be generalized to arbitrary finitely generated varieties of algebras. This generalization is still open (cf. [START_REF] Gillibert | From objects to diagrams for ranges of functors[END_REF]Problem 3]).

A variety V of algebras is strongly congruence-proper if every finite (∨, 0)-semilattice has, up to isomorphism, only finitely many liftings in V and every such lifting is finite (cf. [START_REF] Gillibert | From objects to diagrams for ranges of functors[END_REF]). As a consequence of [START_REF] Freese | Commutator Theory for Congruence Modular Varieties[END_REF]Theorem 10.16], a finitely generated variety of congruence-modular algebras, with finite similarity type, is strongly congruenceproper. As observed in [6, Section 4-10], it follows from [START_REF] Hobby | The Structure of Finite Algebras[END_REF]Theorem 14.6] that a finitely generated variety V of algebras, with finite similarity type, that omits tame congruence theory types 1 and 5, is strongly congruence-proper. This, in turn, occurs in case V satisfies a nontrivial congruence identity (cf. [START_REF] Hobby | The Structure of Finite Algebras[END_REF]Theorem 9.18]). In particular, this holds for varieties of lattices, groups, modules (over finite rings), rings.

Theorem 1.1 is generalized in [START_REF] Gillibert | From objects to diagrams for ranges of functors[END_REF] to strongly congruence-proper varieties of algebras. The aim of the present paper is to improve the bound from < ℵ ω to ≤ ℵ 2 . This solves, in particular, the generalization of the critical point conjecture to the congruence-modular case.

Our main result is the following.

Theorem 5.1. Let V and W be locally finite varieties of algebras. Assume that for each finite algebra A ∈ V there are, up to isomorphism, only finitely many B ∈ W such that Con c A ∼ = Con c B, and every such B is finite. Then either crit(V; W) ≤ ℵ 2 or Con c V ⊆ Con c W.

In particular, the theorem applies to the case where W is strongly congruenceproper (cf. Corollary 5.2). The bound ℵ 2 is optimal since there are finitely generated varieties of lattices (hence strongly congruence-proper) with critical point ℵ 2 . While Theorem 1.2 requires no assumption of either variety V or W be locally finite, we need that assumption for both V and W in the statement of Theorem 5.1. Not every (∨, 0)-semilattice is isomorphic to Con c A for a locally finite algebra (cf. [START_REF] Kearnes | Congruence lattices of locally finite algebras[END_REF]): this suggests that there is still way to go.

In Section 6, we show how to extend our main result from varieties of algebras to quasivarieties of first-order structures (for which a most notable example is given by quasivarieties of graphs, see [START_REF] Gorbunov | Algebraic Theory of Quasivarieties[END_REF]). Theorem 5.1 extends, mutatis mutandis, to Theorem 6.1, by assuming finiteness of the set of relation symbols and changing congruences to relative congruences.

Basic Concepts

We denote by dom f the domain of a function f . Given sets X and Y we denote by X -Y = {x ∈ X | x ∈ Y } the set-theoretical difference of X and Y , and by X Y the set of all maps f : Y → X. For a variety V of algebras and a set X we denote by Fr V (X) the free algebra on X in V.

We denote by 0 (resp., 1) the least (resp., largest) element of a poset if it exists. We denote by 2 = {0, 1} the two-element Boolean algebra. We only consider nonempty algebras. We denote by 0 A the smallest congruence of an algebra A. A (∨, 0)-homomorphism ϕ : S → T separates zero if ϕ(x) = 0 implies that x = 0 for each x ∈ S. Notice that a morphism f of algebras is an embedding if and only if Con c f separates zero. In particular if f and g are morphisms of algebras and there is a natural equivalence Con c f ∼ = Con c g, then f is an embedding if and only if g is an embedding.

Let x < y in a poset P , we write x ≺ y, or, equivalently, y ≻ x, if there is no t ∈ P with x < t < y. Assume that P has 0. An atom of P is an element p ∈ P such that p ≻ 0. We denote by At P the set of all atoms of P .

Let X be a subset of a poset P . We denote

P ↓ X = {p ∈ P | (∃x ∈ X)(p ≤ x)} , P ↑ X = {p ∈ P | (∃x ∈ X)(x ≤ p)} , P ⇑ X = {p ∈ P | (∀x ∈ X)(x ≤ p)} .
A subset Q of P is a lower subset of P (resp., upper subset of P ) if

Q = P ↓ Q (resp., Q = P ↑ Q).
An upper subset Q of P is finitely generated if Q = P ↑ X for some finite subset X of P . An ideal of a poset P is a lower subset I of P such that for all x, y ∈ I there is z ∈ I such that z ≥ x and z ≥ y. An ideal I of P is principal if I = P ↓ {x} for some x ∈ P . We say that P is lower finite if every principal ideal of P is finite. Let θ be a congruence of an algebra A. For an element a of A, we denote by a/θ the equivalence class of a. For a subset X of A we set X/θ = {x/θ | x ∈ X}. If X is a subalgebra of A then X/θ is a subalgebra of A/θ, moreover θ ∩ X 2 is a congruence of X. We shall often identify X/(θ ∩ X 2 ) and X/θ. Given a morphism f : A → B of algebras, the kernel of f is ker

f = {(x, y) ∈ A 2 | f (x) = f (y)}.
Notice that ker f is a congruence of A.

Given algebras A and B and ϕ : Con c A → Con c B a (∨, 0)-homomorphism, we identify ϕ with its natural extension Con A → Con B. That is, given θ ∈ Con A, we set ϕ(θ) = {ϕ(α) | α ∈ Con c A and α ≤ θ}.

Cardinals are initial ordinals, in particular a cardinal is identified with a set. We denote by P(X) the set of all subsets of X. For a cardinal κ, we put

[X] κ = {Y ∈ P(X) | card Y = κ}.
By a diagram in a category S, we mean a functor from a poset, viewed as a category in the usual way (i.e., there is an arrow from p to q iff p ≤ q, and then the arrow is unique), to S. Hence a P -indexed diagram in S is identified with a system (S p , σ q p | p ≤ q in P ) such as σ q p : S p → S q in S, σ p p = id Sp , and σ r p = σ r q • σ q p , for all p ≤ q ≤ r in P .

For an object S of S, the comma category, denoted by S↓S, is the category whose objects are the morphisms f : A → S where A is an object of S and the morphisms from f : A → S to g : B → S are the morphisms h :

A → B of S such that g • h = f .
Denote epimorphisms by f : A ։ B. Given morphisms f : A → B and g : A → C of algebras, we say that f factors through g if there exists h : C → B such that f = h • g. If g is an epimorphism then the map h is unique.

The condensate construction

The proof of the dichotomy result (cf. Theorem 5.1) relies on the condensate construction introduced in [6, Section 3-1]and the Armature Lemma (cf. [START_REF] Gillibert | From objects to diagrams for ranges of functors[END_REF]). In order to ease the understanding of certain crucial parts of our paper, we shall recall the main lines of that construction. The required notions are introduced formally in [6, Chapter 2]. From Section 4 on, the reader can safely forget most of the notations and definitions introduced in Section 3, but should keep in mind the crucial Lemma 3.11, which requires the notion of a norm-covering U of a poset P . Given a P -indexed diagram S in a variety V of algebras, we shall recall the construction of the algebra F(U ) ⊗ S in V and the morphism π U u ⊗ S :

F(U ) ⊗ S → S ∂u , for each u ∈ U .
The following notation is introduced in [6, Section 2-1].

Notation 3.1. Let X be a subset of a poset P , we denote by ▽X the set of all minimal elements of P ⇑ X.

The following definition is given in [6, Section 2-1].

Definition 3.2. A subset X of a poset P is ▽-closed if ▽Y ⊆ X for every finite subset Y of X. The ▽-closure of a subset X of P is the least ▽-closed subset of P containing X.
We say that P is supported if P ⇑ X is a finitely generated upper subset of P and the ▽-closure of X is finite, for every finite (possibly empty) subset X of P .

Notice that the definition of a supported poset is equivalent to the one given in A sharp ideal of (U, ∂) is an ideal u of U such that {∂x | x ∈ u} has a largest element; we denote this element by ∂u. We denote by Id s (U, ∂) the set of all sharp ideals of (U, ∂).

Notice that this definition of a norm-covering is slightly stronger than [6, Section 2-6]. However, this does not affect the definition of an ℵ 0 -lifter (cf. [6, Section 3-2]), as in that case we require the norm-covering to be supported. Remark 3.4. In the context of Definition 3.3, every principal ideal is sharp. The converse does not hold as a rule. However, in the present paper, we shall only consider norm-coverings for which every sharp ideal is principal, in which case we can identify sharp ideals of (U, ∂) with elements of U .

The following definition comes from [6, Section 2-2]. Definition 3.5. Let P be a poset. A P -scaled Boolean algebra A is a Boolean algebra A, together with a family (A (p) | p ∈ P ) of ideals of A, such that:

(1) A = (A (p) | p ∈ P );

(2) A (p) ∩ A (q) = (A (r) | r ≥ p, q in P ), for all p, q ∈ P ; where joins are taken in the lattice of ideals of A. As an immediate consequence the assignment p → A (p) is order-reversing.

A morphism f : A → B of P -scaled Boolean algebras is a morphism f : A → B of Boolean algebras such that f (A (p) ) ⊆ B (p) , for all p ∈ P . We denote by Bool P the category of P -scaled Boolean algebras with morphisms of P -scaled Boolean algebras.

The following definition comes from [6, Section 2-4]. Definition 3.6. A P -scaled Boolean algebra A is compact if A is finite and, for each atom a of A, there is a largest p ∈ P such that a ∈ A (p) . We set |a| = p, for this p.

The finitely presented (in the categorical sense) objects in the category Bool P are exactly the compact P -scaled Boolean algebras (cf. [6, Section 2-4]). Every Pscaled Boolean algebra is a small directed colimit of compact P -scaled Boolean algebras (cf. [6, Section 2-4]).

The following examples of compact P -scaled Boolean algebras appear in [6, Section 2-6]. They will be used in the proof of Lemma 3.11. Notation 3.7. Given p, q in a poset P we put

2[p] (q) = {0, 1}, if q ≤ p {0}, otherwise.
The structure 2[p] = (2, (2[p] (q) | q ∈ P )) is a P -scaled Boolean algebra, for each p ∈ P . Moreover, given p ≤ q, the identity map on {0, 1} defines a morphism of P -scaled Boolean algebras from 2[p] to 2[q]; we denote this morphism by ε q p . We summarize here another family of P -scaled Boolean algebras, constructed in [6, Section 2-6]. For the sake of simplicity we give the notations only in the cases that we need. Notations 3.8. Let (U, ∂) be a norm-covering of a poset P . The Boolean algebra F(U ), defined in [6, Section 2-6], is the Boolean algebra defined by generators u (or u U in case U needs to be specified), for u ∈ U , and the relations:

v ≤ u, for all u ≤ v in U . u ∧ v = ( w | w ∈ ▽{u, v}), for all u, v in U . 1 = ( u | u minimal element of U ).
We denote by F(U ) (p) the ideal of F(U ) generated by { u | u ∈ U and p ≤ ∂u}.

Then

F(U ) = (F(U ), (F(U ) (p) | p ∈ P )) is a P -scaled Boolean algebra (cf. [6, Section 2-6]). Given a ▽-closed subset V of U , we denote by f U V : F(V ) → F(U ) the unique morphism of Boolean algebras such that f U V ( u V ) = u U for all u ∈ V . Moreover, f U V is a morphism of P -scaled Boolean algebras from F(V ) to F(U ) (cf. [6, Section 2- 6]).
Given u ∈ U , we denote by π U u : F(U ) → 2 the unique morphism of Boolean algebras such that

π U u ( v) = 1, if v ≤ u 0, otherwise , for all v ∈ U .
Then π U u defines a morphism of P -scaled Boolean algebras from F(U ) to 2[∂u] (cf. [6, Section 2-6]).

The following construction of condensates appears in [6, Section 3-1]. Notations 3.9. Let V be a variety of algebras, let P be a poset, and let S = (S p , σ q p | p ≤ q in P ) be a P -indexed diagram in V. Given a compact P -scaled Boolean algebra A, we put

A ⊗ S = (S |u| | u ∈ At A),
with canonical projections δ u A : A ⊗ S → S |u| , for all u ∈ At A. Let ϕ : A → B be a morphism of compact P -scaled Boolean algebras. Given an atom v ∈ B, we denote by v ϕ the unique atom u ∈ A such that v ≤ ϕ(u). We define ϕ⊗ S as the unique morphism from A⊗ S to B⊗ S such that

δ v B •(ϕ⊗ S) = σ |v| |v ϕ | •δ v ϕ A for each atom v of B.
This construction defines a functor -⊗ S from the full subcategory of compact P -scaled Boolean algebras of Bool P to V. It it proved in [6, Section 1-4] that this functor can be extended (uniquely up to isomorphism) to the whole category Bool P (cf. [6, Section 3-1]). We denote by -⊗ S this extension.

An object of the form A ⊗ S, for a P -scaled Boolean algebra A, is called a condensate of S.

The construction of ⊗ implies the following lemma (cf. [6, Section 3-1]). Lemma 3.10. Let S = (S p , σ q p | p ≤ q in P ) be a P -indexed diagram in a variety of algebras V. The following equalities hold.

(1) 2[p] ⊗ S = S p , for all p ∈ P .

(2) ε q p ⊗ S = σ q p , for all p ≤ q in P . The following lemma expresses that a condensate of a diagram contains copies of the algebras in the diagram. Lemma 3.11. Let U be a norm-covering of a poset P . Assume that both U and P have a least element and that ∂0 = 0. Let S = (S p , σ q p | p ≤ q in P ) be a diagram in V. There is a morphism

d 0 : S 0 → F(U ) ⊗ S such that (π U u ⊗ S) • d 0 = σ ∂u 0 for each u ∈ U . Let u ∈ U -{0}.
There is a morphism d u : S 0 × S ∂u → F(U ) ⊗ S such that, denoting t 0 : S 0 × S ∂u → S 0 and t 1 : S 0 × S ∂u → S ∂u the canonical projections, the following equality holds

(π U v ⊗ S) • d u = σ ∂v 0 • t 0 , if u ≤ v σ ∂v ∂u • t 1 , if u ≤ v , for each v ∈ U .
Proof. Notice that F({0}) = {0, 1} is the two-element Boolean algebra. Moreover, given p > 0 in P , F({0}) (p) = {0} and F({0}

) (0) = {0, 1}. Hence F({0}) = 2[0], thus it follows from Lemma 3.10(1) that F({0}) ⊗ S = S 0 . Notice that {0} is a ▽-closed subset of U . Put d 0 = f U {0} ⊗ S (cf. Notations 3.8). Given u ∈ U the following equalities hold (π U u ⊗ S) • d 0 = (π U u ⊗ S) • (f U {0} ⊗ S) = (π U u • f U {0} ) ⊗ S, as -⊗ S is a functor = ε ∂u 0 ⊗ S, as π U u • f U {0} = ε ∂u 0 = σ ∂u 0 , by Lemma 3.10(2).
Let u ∈ U -{0}. Notice that F({0, u}) = {0, ¬ u, u, 1}, with 0 = 1, is the four-element Boolean algebra. Moreover Notice that {0, u} is a ▽-closed subset of U . The following equality holds

F({0, u}) (p) =      {0}, if p ≤ ∂u {0, u}, if 0 < p ≤ ∂u {0, ¬ u, u, 1}, if p = 0
π U v • f U {0,u} = ε ∂v 0 • π {0,u} 0 , if u ≤ v ε ∂v ∂u • π {0,u} u , if u ≤ v , for each v ∈ U .
Therefore, the map d u = f U {0,u} ⊗ S satisfies the required conditions.

Lifting poset-indexed diagrams

In this section we fix varieties V and W of algebras, an infinite cardinal κ, an algebra G ∈ W, and an isomorphism ξ : Con c Fr V (κ) → Con c G.

Given a finite algebra A, the following lemma expresses that there is a large family of quotients Fr V (κ) ։ A such that the corresponding quotients of G have all the same cardinality. For a function t from a subset of κ to A, we denote by t : Fr V (κ) → A the unique homomorphism extending t and sending every element of κdom(t) to c. For each X 0 ∈ [κ] κ , there are X ∈ [X 0 ] κ and an integer n such that the following statements hold:

(1) Let t : X ։ A. Then card G/ξ(ker t) ≤ n.

(2) For each Y ∈ [X] κ there exists t : Y ։ A such that card G/ξ(ker t) = n.

Proof. Let t : κ։A. Notice that Con c G/ξ(ker t) ∼ = Con c Fr V (κ)/ ker t ∼ = Con c A, so card G/ξ(ker t) ≤ m , for each t : κ ։ A.

Given X ∈ [X 0 ] κ we set n X = max{card G/ξ(ker t) | t : X ։ A}, note that the maximum exists and is ≤ m by (4.1). Moreover

n Y ≤ n X , for all Y ⊆ X in [X 0 ] κ . (4.2) Fix X ∈ [X 0 ] κ such that n X is smallest possible. It follows from the definition of n X that (1) holds for n = n X . Let Y ∈ [X] κ .
From the minimality of n X we have n Y ≥ n X , the equality n Y = n X follows from (4.2). Therefore there is t :

Y ։ A such that card G/ξ(ker t) = n Y = n X . That is (2) holds for n = n X .
Notice that given X ∈ [κ] κ that satisfies the conditions (1) and (2) of Lemma 4.1, then every X ′ ∈ [X] κ also satisfies those conditions. Hence an easy induction argument yields the following lemma. Lemma 4.2. Let (A i ) i∈I be a finite family of finite algebras of V, fix c i ∈ A i for each i ∈ I. Assume that there is an integer m such that card B ≤ m for each algebra B ∈ W with Con c B ∼ = Con c A i for some i ∈ I. For a function t from a subset of κ to A i , we denote by t (i) : Fr V (κ) → A i the unique homomorphism extending t and sending each element of κdom(t) to c i .

Then for each X 0 ∈ [κ] κ , there are X ∈ [X 0 ] κ and a family (n i ) i∈I of integers such that the following statements hold:

(1) Let t : X ։ A i . Then card G/ξ(ker t (i) ) ≤ n i ;

(2) For each Y ∈ [X] κ there exists t : Y ։A i such that card G/ξ(ker t (i) ) = n i ;

for each i ∈ I.

Lemma 4.3. In the context of Lemma 4.2, if there is an isomorphism α :

A i → A j such that α(c i ) = c j , then n i = n j .
Proof. It follows from Lemma 4.2(2), applied to Y = X, that there exists t :

X ։A i such that card G/ξ(ker t (i) ) = n i . Put s = α • t. Notice that s (j) = α • (t (i)
), so ker s (j) = ker t (i) , thus n i = card G/ξ(ker t (i) ) = card G/ξ(ker s (j) ) ≤ n j . With a similar argument we obtain n j ≤ n i .

The following lemma illustrates that a natural transformation can be factored through a natural equivalence.

Lemma 4.4. Let I be a poset, let G = (G i , g j i | i ≤ j in I) be a diagram of algebras, let D = (D i , d j i | i ≤ j in I) be a diagram of (∨, 0)-semilattices, and let χ = (χ i | i ∈ I) : Con c • G → D be a natural transformation. Let θ i be a congruence of G i , denote by p i : G i ։ G i /θ i
the canonical projection, for each i ∈ I. We assume that each χ i factors through Con c p i to an isomorphism. Then g j i induces a morphism β j i : G i /θ i → G j /θ j for all i ≤ j in I. Moreover, B = (G i /θ i , β j i | i ≤ j in I) is a diagram of algebras and χ induces a natural isomorphism from Con c • G to D.

Proof. For i ∈ I, we denote by τ i : Con c (G i /θ i ) → D i the isomorphism induced by χ i , that is, τ i is the unique map such that χ i = τ i • Con c p i . Let i ≤ j in I, let α ⊆ θ i be a compact congruence of B i . The following equalities hold

j • Con c g j i )(α) = (d j i • χ i )(α), as χ is a natural transformation = (d j i • τ i • Con c p i )(α) = (d j i • τ i )(0 Gi/θi ), as α ⊆ θ i = ker p i = 0. As χ j = τ j • Con c p j , it follows that (Con c p j )((Con c g j i )(α)) = 0 Gj/θj , therefore (Con c g j i )(α) ⊆ ker p j = θ j . Let (x, y) ∈ θ i . Considering α = Θ Gi (x, y) ⊆ θ i , we see that (g j i (x), g j i (y)) ∈ (Con c g j i )(Θ Gi (x, y)) ⊆ θ j . Therefore g j i induces a homomorphism β j i : G i /θ i → G j /θ j . It is easy to check that B = (G i /θ i , β j i | i ≤ j in I) is a diagram of algebras. By construction β j i • p i = p j • g j i ,
hence the square (3) of the diagram in Figure 1 commutes. As χ is a natural transformation, the square (1) of the diagram in Figure 1 commutes. By definition of τ i and τ j , both triangles (2) and ( 4) of the diagram in Figure 1 An ℵ 0 -lifter of a poset P is a norm-covering (U, ∂) of P , endowed with a set of sharp ideals U ⊆ Id s (U, ∂) and for which the map ∂ : U → P has a collection of right inverses satisfying certain infinite combinatorial properties (cf. [START_REF] Gillibert | From objects to diagrams for ranges of functors[END_REF]). This property implies, in particular, that ∂ is surjective. Lemma 4.5. Let I be a finite lattice, let A = (A i , α j i | i ≤ j in I) be a diagram in V. Let (U, U ) be an ℵ 0 -lifter of I such that card U ≤ κ. We assume that the following statements hold.

commute. D j (2) Con c G j χj O O Conc pj / / Con c G j /θ j τj q q (1) (3)
Con c G i Conc g j i O O Conc pi / / χi Con c G i /θ i τi m m Conc β j i O O (4)
(1) Every element of U is a principal ideal.

(2) The poset U is lower finite and has a least element.

(3) ∂u ≻ 0 implies that u ≻ 0 for each u ∈ U . (4) There is an integer m such that card B ≤ m for each algebra B ∈ W with Con c B ∼ = Con c A i for some i ∈ I. (5) The algebra A 0 is generated by one element. [START_REF] Gillibert | From objects to diagrams for ranges of functors[END_REF] The algebra G is locally finite.

Then there exists a lifting B = (B i ,

β j i | i ≤ j in I) of Con c • A in W, such that for all 0 ≺ i ≤ j in I, if α j
i is an isomorphism, then β j i is an isomorphism.

Proof. From the surjectivity of ∂ it follows that ∂0 = 0.

Denote by c 0 a generator of A 0 and put c i = α i 0 (c 0 ) for all i ∈ I. For a function t from a subset of κ to A i , we denote by t (i) : Fr V (κ) → A i the unique homomorphism extending t and sending each element of κdom(t) to c i . By Lemma 4.2, there are X ∈ [κ] κ and a family (n i ) i∈I of integers such that the following statements hold: [START_REF] Goodearl | Tensor products of dimension groups and K 0 of unitregular rings[END_REF] Let i ∈ I and let t :

X ։ A i . Then card G/ξ(ker t (i) ) ≤ n i . (8) Let Y ∈ [X] κ , let i ∈ I.
There exists t : Y ։A i , such that card G/ξ(ker t (i) ) = n i . Moreover, we can assume that κ -X is not empty.

Claim 1. There exists a morphism f : Fr V (κ) → F(U ) ⊗ A such that the following statements hold: [START_REF] Grätzer | Lattice Theory: Foundation[END_REF] The morphism

(π U u ⊗ A) • f is surjective, for each u ∈ U . (10) Let 0 ≺ u ≤ v in U . If α ∂v ∂u is an isomorphism, then card G/ξ(ker((π U u ⊗ A) • f )) ≥ card G/ξ(ker((π U v ⊗ A) • f )) . Proof of Claim. Set k u = π U u ⊗ A, for each u ∈ U . Put U * = U -{0}. Fix morphisms d 0 : A 0 → F(U ) ⊗ A and d u : A 0 × A ∂u → F(U ) ⊗ A, for u ∈ U * ,
as in Lemma 3.11 (with S replaced by A). In particular

k u • d 0 (a) = α ∂u 0 (a)
, for all u ∈ U and all a ∈ A 0 .

k v • d u (a, b) = α ∂v 0 (a), for all u ≤ v in U , a ∈ A 0 , and b ∈ A ∂u . (4.3)
k v • d u (a, b) = α ∂v ∂u (b), for all 0 < u ≤ v in U , a ∈ A 0 , and b ∈ A ∂u . (4.5) As card U * ≤ κ = card X, there is a partition (X u ) u∈U * of X such that card X u = κ for each u ∈ U . Put X 0 = κ -X, so (X u ) u∈U is a partition of κ. Denote by f 0 : X 0 → F(U ) ⊗ A, x → d 0 (c 0 ) the constant map. It follows from (4.3) that k v (f 0 (x)) = k v (d 0 (c 0 )) = α ∂v 0 (c 0 ) = c ∂v (4.4)
, for all x ∈ X 0 and all v ∈ U . Thus the following equality holds.

k v • f 0 = c ∂v , the constant map, for all v ∈ U . (4.6)
Let u ∈ U * . As X u ⊆ X and card X u = κ, it follows from (8) that there exists t u : X u ։ A ∂u such that the following equality holds card G/ξ(ker t (∂u)

u ) = n ∂u . (4.7)
Put

f u : X u → F(U ) ⊗ A, x → d u (c 0 , t u (x)). From (4.5) we obtain k u (f u (x)) = α ∂u ∂u (t u (x)) = t u (x), for all x ∈ X u . Hence k u • f u = t u , for each u ∈ U * . ( 4.8)
Similarly, (4.4) implies that

k v • f u = c ∂v , the constant map, for each u ≤ v in U . ( 4.9)
The family (X u ) u∈U is a partition of κ, so there is a (unique) morphism of algebras

f : Fr V (κ) → F(U ) ⊗ A that extends f u for each u ∈ U . Let u ∈ U * . As t u is surjective it follows from (4.8) that k u • f is surjective. As X 0 is not empty, we see from (4.6) that the image of k 0 • f contains (as an element) c 0 , which is a generator of A 0 , so k 0 • f is surjective. Therefore, f satisfies (9). Let u ≻ 0 in U . Let x ∈ κ, let v in U be such that x ∈ X v . If v ≤ u, then it follows from (4.9) that k u • f (x) = c ∂u = t (∂u) u (x). If v < u, then v = 0, hence Con c (G/θ ′ u ) ξ ′ u / / Con c Fr V (κ)/θ u Conc hu { { (2)
Con c (G u /θ ′ u ) Conc g ′ u (1) Con c G Conc pu O O ξ -1 / / Con c Fr V (κ) Conc ψu Conc qu O O (4) (3)
Con c G u Conc p ′ u e e ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ ϕu =Conc gu O O ρu•ϕu / / Con c A ∂u Figure 2. The map ρ u • ϕ u factors through Con c p ′ u As G u is finite, S u = Con c G u is finite.
By applying the Armature Lemma (cf. [6, Section 3-2]) to the functor Con c on V, we obtain that there is an isotone section σ : I ֒→ U such that the family (ρ

σ(i) • ϕ σ(i) | i ∈ I) is a natural transformation from (S σ(i) , ϕ σ(j) σ(i) | i ≤ j in I) to Con c • A. It follows from Claim 2 that the map ρ σ(i) • ϕ σ(i) induces an isomorphism τ i : Con c G σ(i) /θ ′ σ(i) → Con c A i for each i ∈ I. Put B i = G σ(i) /θ ′ σ(i)
. It follows from Lemma 4.4 that g σ(j) σ(i) induces a morphism β j i : B i → B j for all i ≤ j in I. This defines a diagram B = (B i , β j i | i ≤ j in I), and τ is a natural equivalence from Con c • B to Con c • A.

Let 0 ≺ i ≤ j such that α j i is an isomorphism. In particular, α j i is an embedding, thus β j i is also an embedding. As ∂σ(i) = i ≻ 0, it follows from Assumption (3) that σ(i) ≻ 0. Thus, from Assumption [START_REF] Hobby | The Structure of Finite Algebras[END_REF]

(cf. Claim 1) we obtain that card G/θ ′ σ(i) ≥ card G/θ ′ σ(j) . However B i = G σ(i) /θ ′ σ(i) = G/θ ′ σ(i) , similarly B j = G/θ ′ σ(j)
, so it follows that card B i ≥ card B j . However, β j i : B i → B j is an embedding, therefore β j i is an isomorphism.

The following lemma is proved in [START_REF] Gillibert | Critical points between varieties generated by subspace lattices of vector spaces[END_REF]Lemma 4.11]. We remind the reader that an ℵ 0 -compatible norm-covering, as defined in [START_REF] Gillibert | Critical points between varieties generated by subspace lattices of vector spaces[END_REF]Definition 4.4], is nothing else as an ℵ 0 -lifter. Lemma 4.6. Let X be a finite set and set P = {P ⊆ X | card P ≤ 2 or P = X}. Define U as the set of all functions from a subset of X to ℵ 2 , partially ordered by inclusion. Let

∂ : U → P u → ∂u = dom u if card(dom u) ≤ 2, X otherwise.
Denote by U the set of all principal ideals of U . Then (U, U ) is an ℵ 0 -lifter of P. Moreover, card U = ℵ 2 .

Remark 4.7. In the context of Lemma 4.6, the poset U is lower finite and has a smallest element. Moreover, ∂u ≻ 0 implies that dom u = {x} for some x ∈ X, hence u ≻ 0. Hence the truncated Boolean algebra P has an ℵ 0 -lifter that satisfies the conditions (1), [START_REF] Freese | Commutator Theory for Congruence Modular Varieties[END_REF], and (3) of Lemma 4.5.

The following lemma expresses that a diagram of (∨, 0)-semilattices with a lifting in V has a lifting in W.

Q P = α ∨Q ∨P : A ′ P → A ′ Q for all P ⊆ Q in P. This defines a diagram A ′ = (A ′ P , f Q P | P ⊆ Q in P).
It follows from Lemma 4.6 and Remark R:normcov that there is an ℵ 0 -lifter (U, U ) of P, such that card U = ℵ 2 and the following statements hold:

(1) Every element of U is a principal ideal.

(2) The poset U is lower finite and has a smallest element.

(3) ∂u ≻ 0 implies that u ≻ 0 for each u ∈ U . The assumptions of Corollary 4.8 imply that:

(4) There is an integer m such that card B ≤ m for each algebra B ∈ W with Con c B ∼ = Con c A i for some i ∈ I. Moreover, by construction, A ′ ∅ is generated by one element and we have assumed that G is locally finite. It follows from Lemma 4.5 that there exists a lifting B = (B P ,

β Q P | P ⊆ Q in P) of Con c • A ′ in W, such that for all ∅ ≺ P ⊆ Q in P, if f Q P is an isomorphism, then β Q P is an isomorphism. Put C i = B {i} , for all i ∈ I. Let i ≤ j in I. Notice that f {i,j} {j} = α j j = id Aj is an isomorphism, hence β {i,j}
{j} is an isomorphism. The map

g j i = β {i,j} {j} -1 • β {i,j} {i} is a morphism from C i to C j .
The following equalities hold

β I {j} • g j i = β I {i,j} • β {i,j} {j} • β {i,j} {j} -1 • β {i,j} {i} = β I {i,j} • β {i,j} {i} = β I {i} . Hence β I {j} • g j i = β I {i} , for all i ≤ j in I . ( 4.11)
Let i ≤ j ≤ k, it follows from (4.11) that

β I {k} • g k j • g j i = β I {j} • g j i = β I {i} = β I {k} • g k i .
Moreover, Hence (τ {i} ) i∈I is a natural equivalence from Con c • A to Con c • C.

f I {k} = α 1 k is an embedding, hence β I {k} is an embedding, thus g k j •g j i = g k i . Therefore, (C i , g j i | i ≤ j in I)

Critical points

We can now prove the main result of this paper.

Theorem 5.1. Let V and W be locally finite varieties of algebras. Assume that for each finite algebra A ∈ V there are, up to isomorphism, only finitely many B ∈ W such that Con c A ∼ = Con c B, and every such B is finite. Then either crit(

V; W) ≤ ℵ 2 or Con c V ⊆ Con c W.
Proof. Assume that crit(V; W) > ℵ 2 . The algebra Fr V (ℵ 2 ) is locally finite, so card Fr V (ℵ 2 ) ≤ ℵ 2 , hence card Con c Fr V (ℵ 2 ) ≤ ℵ 2 . There are G ∈ W and an isomorphism ξ : Con c Fr V (ℵ 2 ) → Con c G.

The remaining of the proof is similar to the Dichotomy Theorem of [6, Section 4-9], using Corollary 4.8.

Let A ∈ V, let a ∈ A. Denote by P the set of all finite subalgebras of A containing a. Set A p = p, denote by α q p : A p → A q , and by α p : A p → A the inclusion maps, for all p ≤ q in P . Put A = (A p , α q p | p ≤ q in P ). As V is locally finite, the poset P is a (∨, 0)-semilattice and (A, α p | p ∈ P ) is a colimit cocone of A.

Let I be a finite (∨, 0)-subsemilattice of P , by Corollary 4.8 the diagram Con c • A↾ I has a lifting in W, it follows from the Compactness Lemma of [6, Section 4-9] that Con

c • A has a lifting B in W. Fix B ∈ W a colimit of B. The following isomor- phisms hold Con c A ∼ = Con c (lim -→ A), as A is a colimit of A ∼ = lim -→ (Con c • A), as Con c preserves directed colimits ∼ = lim -→ (Con c • B), as Con c • A and Con c • B are naturally isomorphic ∼ = Con c (lim -→ B), as Con c preserves directed colimits ∼ = Con c B, as B is a colimit of B. Hence Con c A has a lifting in W for each A ∈ V, that is, Con c V ⊆ Con c W.
If W is strongly congruence-proper, then the condition of Theorem 5.1 is satisfied, we deduce the following result.

Corollary 5.2. Let V and W be locally finite varieties of algebras. If W is strongly congruence-proper, then either crit(V; W) ≤ ℵ 2 or Con c V ⊆ Con c W.

By using the results of [START_REF] Hobby | The Structure of Finite Algebras[END_REF], we observed, in [START_REF] Gillibert | From objects to diagrams for ranges of functors[END_REF], that a finitely generated variety of algebras that satisfies a nontrivial congruence identity is strongly congruence-proper. Therefore, the following result is a consequence of Corollary 5.2.

Corollary 5.3. Let V be a locally finite variety of algebras. Let W be a finitely generated variety of algebras, with finite similarity type. If W satisfies a nontrivial congruence identity, then either crit(V; W) ≤ ℵ 2 or Con c V ⊆ Con c W.

Extension to quasivarieties of first-order structures

We conclude the paper with a word on quasivarieties of first-order structures, as considered in [START_REF] Gorbunov | Algebraic Theory of Quasivarieties[END_REF]. We briefly recall the basic definitions. A class of first-order structures on a first-order language L is a quasivariety if it is closed under substructures, direct products, and directed colimits (within the class of all models for L ). This notion is quite robust and has many equivalent forms, see [START_REF] Gorbunov | Algebraic Theory of Quasivarieties[END_REF] for details. In [6, Section 4-1], we define a congruence of a first-order structure A as an equivalence relation on (the universe of) A, augmented by a family of subsets of finite powers of A indexed by the set of all relation symbols of L , satisfying certain compatibility conditions. (In particular, if there are no relation symbols, then a congruence is an equivalence relation.) This definition is equivalent to the one introduced in [START_REF] Gorbunov | Algebraic Theory of Quasivarieties[END_REF]. Congruences of a first-order structure A are in one-to-one correspondence with surjective homomorphisms with domain A up to isomorphism. A most important class of examples, extensively considered in [START_REF] Gorbunov | Algebraic Theory of Quasivarieties[END_REF], is given by graphs.

Unlike varieties, quasivarieties may not be closed under homomorphic images. Thus the relevant concept of congruence, for a member A of a quasivariety V, is often modified by considering only the V-congruences (or congruences relative to V): by definition, a congruence θ of A is a V-congruence if the quotient A/θ is a member of V. The set Con V A of all V-congruences of A, partially ordered by inclusion, is still an algebraic lattice, and we denote by Con V c A its (∨, 0)-semilattice of compact elements. Furthermore, we define Con c,r V (where the letter "r" stands for "relative") as the class of all isomorphic copies of Con V c A for A ∈ V. The relative critical point crit r (V; W) between quasivarieties V and W of firstorder structures is defined as the least cardinality of a member of the difference (Con c,r V) -(Con c,r W), if Con c,r V ⊆ Con c,r W; and ∞ otherwise. Our main result, Theorem 5.1, extends mutatis mutandis to quasivarieties of first-order structures and relative congruence lattices. Aside from a few additional arguments dealing with the relation symbols, very similar to those dealing with equality for varieties of algebras, the proofs are virtually the same. An important point is the finiteness assumption on the sets of relation symbols in the languages of both V and W, which is essentially required in order to ensure that Con A is finite whenever (the universe of) A is finite. We thus improve the estimate of the Dichotomy Theorem stated in [6, Section 4-9], namely crit r (V; W) < ℵ ω , to crit r (V; W) ≤ ℵ 2 , moreover under a slightly weaker assumption. Theorem 6.1. Let V and W be locally finite quasivarieties of first-order structures, in first-order languages with only finitely many relation symbols. Assume that for each finite A ∈ V there are, up to isomorphism, only finitely many B ∈ W such that Con V c A ∼ = Con W c B, and every such B is finite. Then either crit r (V; W) ≤ ℵ 2 or Con c,r V ⊆ Con c,r W.

It is still not known whether the assumption, stating that for any finite A ∈ V there are only finitely many B ∈ B such that Con V c A ∼ = Con W c B, can be dispensed with. Due to the example in [10, Exercise 14.9(4)], attributed there to C. Shallon, this assumption does not hold as a rule: there is a finitely generated variety of algebras with a proper class of simple members. We also do not know whether the local finiteness assumption, on both quasivarieties V and W, can be dispensed with.

Acknowledgment

I thank Friedrich Wehrung, for many constructive remarks and the help to write this paper.

[ 3 ,

 3 Definition 4.1]. The kernels of a supported poset P are the finite ▽-closed subsets of P . The following definition is given in [3, Definition 4.3]. Definition 3.3. A norm-covering of a poset P is a pair (U, ∂) where U is a supported poset and ∂ : U → P is an isotone map.

,

  for all p ∈ P . It follows that | u| = ∂u and |¬ u| = 0, hence F({0, u}) ⊗ S = S 0 × S ∂u . Moreover, π {0,u} 0 is the unique morphism from F({0, u}) to 2[∂0] = 2[0] that sends u to 0, while π {0,u} u is the unique morphism from F({0, u}) to 2[∂u] that sends u to 1. It follows that π {0,u} 0 ⊗ S = t 0 and π {0,u} u ⊗ S = t 1 .

Lemma 4 . 1 .

 41 Let A be a finite algebra of V, fix c ∈ A. Assume that there is an integer m such that card B ≤ m for every algebra B ∈ W with Con c B ∼ = Con c A.

Figure 1 .

 1 Figure 1. The family τ is a natural isomorphism

Corollary 4 . 8 .

 48 Assume that κ = ℵ 2 and that G is locally finite. Let A be a diagram of finite algebras and embeddings in V, indexed by a finite (∨, 0)-semilattice I. Assume that for each algebra A of the diagram A, there is a finite bound on the cardinality of liftings of Con c A in W. Then the diagram Con c • A is liftable in W. Proof. Write A = (A i , α j i | i ≤ j in I) and put P = {P ⊆ I | card P ≤ 2 or P = I}. Let us choose c ∈ A 0 , denote by A ′ ∅ the subalgebra of A 0 generated by c, and put A ′ P = A ∨P for each nonempty P ∈ P. Put f

1 •= Con c β {i,j} {j} - 1 •

 11 is a diagram of algebras in W. Let τ = (τ P ) P ∈P : Con c • A ′ → Con c • B be a natural equivalence. Let i ≤ j in I. As τ is a natural equivalence, Con c β {i,j} {j} • τ {j} = τ {i,j} • Con c f {i,j} {j} . However f {i,j} {j} = α j j = id Aj , hence Con c β {i,j} {j} • τ {j} = τ {i,j} , thus the following equality holds τ {j} = (Con c β {i,j} {j} ) -1 • τ {i,j} . (4.12) Therefore, we obtain (Con c g j i ) • τ {i} = Con c β {i,j} {j} -(Con c β {i,j} {i} ) • τ {i} , by the definition of g j i τ {i,j} • (Con c f {i,j} {i} ), as τ is a natural equivalence = τ {j} • (Con c f {i,j} {i} ), by (4.12) = τ {j} • (Con c α j i ), as f {i,j} {i} = α j i .

This work was partially supported by the institutional grant MSM 0021620839.

 (4.6), therefore (4.7) implies that card G/ξ(ker(k u • f )) = n ∂u , for each u ≻ 0 in U .

(4.10)

It follows from (4.8) that the range of t contains the range of t v , which is equal to ∂v) . Therefore, by [START_REF] Goodearl | Tensor products of dimension groups and K 0 of unitregular rings[END_REF],

However, n ∂v = n ∂u (cf. Lemma 4.3). Thus, by (4.10), we obtain that card G/ξ(ker

We fix a morphism f :

Denote by S the category of all (∨, 0)-semilattices with (∨, 0)-homomorphisms and put S = Con c G. For all u ≤ v in U , set S u = Con c G u , denote by g u : G u ֒→ G and g v u : G u ֒→ G v the inclusion maps, and set ϕ v u = Con c g v u and ϕ u = Con c g u . This defines a diagram ((S u , ϕ u ),

Denote by p

Denote by g ′ u : G u /θ ′ u → G/θ ′ u the morphism induced by the inclusion map g u , so the square (1) of the diagram in Figure 2 commutes. It follows from the choice of G u that g ′ u is an isomorphism. Denote by ξ ′ u : Con c (G/θ ′ u ) → Con c Fr V (κ)/θ u the (∨, 0)-homomorphism induced by ξ -1 . Then the square (2) of the diagram in Figure 2 commutes. As θ ′ u = ξ(θ u ) and ξ -1 is an isomorphism, ξ ′ u is an isomorphism. Denote by q u : Fr V (κ) ։ Fr V (κ)/θ u the canonical projection, for each u ∈ U . As ψ u = h u • q u , the triangle (4) of the diagram in Figure 2 commutes.

Therefore, the diagram in Figure 2 commutes, hence

is an isomorphism, the conclusion follows. Claim 2.


Maxime Dahirel
email: maxime.dahirel@yahoo.fr

Armelle Ansart

Luc Madec

Stage-and weather-dependent dispersal in the brown garden snail Cornu aspersum

Keywords: Dispersal distances, Helix aspersa, Life-history strategies, Natal dispersal, Phenotype-dependent dispersal

Dispersal decisions are often condition-dependent, influenced by the interaction of individual phenotype and environmental conditions. Terrestrial Gastropods are simultaneous hermaphrodites, a reproductive system rarely studied in the context of dispersal. Moreover, the energetic cost of their movement is one of the highest among animals. Despite these features, which make them valuable models to understand the trade-offs between dispersal and other lifehistory traits, their dispersal strategies have been barely explored. We studied the movements of subadults and adults of the brown garden snail Cornu aspersum in a semi-natural 4-patch network, for two months in 2011 (a dry year) and one month in 2012 (a wet year). We assessed the effects of life-history stage (subadult/adult) and weather conditions on dispersal propensity and dispersal speed. Snails were more mobile under humid and warm weather, but nearly all individuals left patches when the relative humidity was close to 100 % in 2012. Because such humidity levels are potentially lethal to C. aspersum, we argue these extreme emigration rates might be an emergency escape response to harmful conditions. Despite a theoretically higher cost of movement, we found that subadults emigrated more, and dispersed faster and further, than adults. Thus, and contrary to what was expected, direct costs of movement do not play the main role in shaping dispersal in C. aspersum. Observed differences between subadults and adults in dispersal behaviour are discussed in the context of intraspecific competition, inbreeding avoidance and relative costs of male and female reproduction.

Introduction

Dispersal, defined as "any movement of individuals or propagules with potential consequences for gene flow across space" [START_REF] Ronce | How does it feel to be like a rolling stone? Ten questions about dispersal evolution[END_REF], plays a key role in many ecological and evolutionary processes (Bowler and Benton 2005;[START_REF] Clobert | Dispersal ecology and evolution[END_REF]. Benefits of dispersal include reduced kin competition, inbreeding avoidance, and coping with environmental stochasticity [START_REF] Starrfelt | The theory of dispersal under multiple influences[END_REF]. It is now widely acknowledged that dispersal is not a species-fixed trait, but can be highly variable both within and among populations [START_REF] Clobert | Dispersal ecology and evolution[END_REF]). One reason is that benefits and costs of dispersal vary across environments, between sexes and between different age classes, leading to context-, sex-or age-related differences in dispersal propensity and dispersal distances (Bowler and Benton 2005;Bitume et al. 2013). Moreover, the effect of environmental conditions is likely to differ between sexes or life-history stages, leading to potentially complex interactions [START_REF] Bowler | Variation in dispersal mortality and dispersal propensity among individuals: the effects of age, sex and resource availability[END_REF]. These aspects have been mostly studied in arthropods, vertebrates, and plants, leading to a strong taxonomic bias in our understanding of dispersal. Thus, consistent information about dispersal is still lacking for a wide part of the range of existing life histories, including some otherwise well-studied groups.

Land snails and slugs (Stylommatophoran gastropods) form a highly diverse group of molluscs, with more than 30,000 species distributed all around the world in a great variety of ecosystems (Barker 2001a). Many species are important crop pests in and outside their native range [START_REF] Barker | Molluscs as crop pests[END_REF], and some introduced species have colonized natural habitats, with negative impacts on native biodiversity [START_REF] Cowie | Snails and slugs[END_REF]. Thus, knowledge of their dispersal capacities is of conservation and economic interest [START_REF] Aubry | Active and passive dispersal of an invading land snail in Mediterranean France[END_REF][START_REF] Honek | Body size and the colonisation of cereal crops by the invasive slug Arion lusitanicus[END_REF]. In terrestrial gastropods, long-distance dispersal and introductions across regions are passive processes, mainly mediated by wind, water and human transport [START_REF] Dörge | The significance of passive transport for dispersal in terrestrial snails (Gastropoda, Pulmonata)[END_REF][START_REF] Aubry | Active and passive dispersal of an invading land snail in Mediterranean France[END_REF]). However, expansion and gene flow at local scales seem to depend mostly on active dispersal [START_REF] Aubry | Active and passive dispersal of an invading land snail in Mediterranean France[END_REF]. Despite that, to date, studies of the active movement capacity of land snails and slugs are rare, and mainly descriptive (see Baur 1993aBaur , 1993b;;[START_REF] Aubry | Active and passive dispersal of an invading land snail in Mediterranean France[END_REF], for notable exceptions). We nonetheless believe that a better understanding of dispersal in land snails and slugs can shed new light on some general questions about the ecology and evolution of dispersal. Indeed, their peculiar life histories may have led to original, yet unstudied, dispersal strategies.

Stylommatophoran gastropods are simultaneous hermaphrodites, with a tendency for protandry in at least some lineages (Barker 2001b). The presence of both male and female reproductive organs in a single individual adds a dimension to dispersal/reproduction trade-offs, which play a key role in the evolution of dispersal [START_REF] Ims | Condition-dependent dispersal[END_REF][START_REF] Danchin | Can non-breeding be a cost of breeding dispersal?[END_REF]. In gonochoric animals, sex-biased dispersal can be selected as a means to avoid inbreeding, by increasing the distance between opposite-sex siblings [START_REF] Perrin | Inbreeding, kinship, and the evolution of natal dispersal[END_REF][START_REF] Costello | Sex-biased natal dispersal and inbreeding avoidance in American black bears as revealed by spatial genetic analyses[END_REF]. Sex-biased dispersal can have profound effects on metapopulation dynamics, for instance by altering the speed of colonisation processes [START_REF] Miller | Sex and stochasticity affect range expansion of experimental invasions[END_REF]. It can also increase the extinction probability of small populations, by increasing the mismatch between the male and female spatial distributions, thus decreasing the chances to find a mate for both the dispersing and the philopatric sex [START_REF] Dale | Female-biased dispersal, low female recruitment, unpaired males, and the extinction of small and isolated bird populations[END_REF][START_REF] Miller | Sex and stochasticity affect range expansion of experimental invasions[END_REF]. Under similar selective pressures, we can expect that hermaphroditic animals would evolve quite different dispersal solutions, with potentially different consequences on population dynamics.

Costs of dispersal have been repeatedly shown to be instrumental in the evolution of this behaviour (Bonte et al. 2012). Costs are of different types (e.g., energetic, mortality or time costs) and can be incurred at all stages of the dispersal process (before emigration, at departure, during transience, and at settlement). These costs usually lead to trade-offs between dispersal and other life-history traits. For instance, in winged insects, increased resource allocation to wings and muscles is typically done at the expense of fecundity [START_REF] Guerra | Evaluating the life-history trade-off between dispersal capability and reproduction in wing dimorphic insects: a meta-analysis[END_REF]. Moreover, actively dispersing individuals are expected to adjust their decisions to settle or continue based on currently perceived costs, affecting the distribution of dispersal distances (Bonte et al. 2012;Bitume et al. 2013). The energetic cost of movement in snails and slugs is one of the highest among animals; they spend about 10 times more energy per meter moved than any other terrestrial animal of similar weight [START_REF] Denny | Locomotion: the cost of gastropod crawling[END_REF]. Moreover, because of their relative slowness, dispersing snails and slugs might be subject to a higher predation risk compared to more mobile animals, increasing the mortality cost of dispersal. However, land snails also carry a protective shell of variable thickness, which limits predation by non-specialists [START_REF] Barker | Natural enemies of terrestrial molluscs[END_REF], as well as decreases the risks associated with body water evaporation under unfavourable conditions [START_REF] Klein-Rollais | Contribution à l'étude de la balance hydrique et de sa régulation chez l'escargot Petit-Gris[END_REF]. These characteristics make this taxon an interesting model to understand how dispersal costs, in particular costs during transience, interact with each other and affect the trade-offs between dispersal and other life-history traits.

Here we investigate the potential changes in dispersal capacity between life-history stages in the brown garden snail Cornu aspersum (Müller) (fam. Helicidae; syn. Helix aspersa Müller).

Native of the Mediterranean area, this relatively large land snail (shell diameter at maturity: 25 to 40 mm) has been spread throughout the world by humans, either voluntarily, for food purposes, or accidentally [START_REF] Guiller | Tracing the invasion of the mediterranean land snail Cornu aspersum aspersum becoming an agricultural and garden pest in areas recently introduced[END_REF]). This species is now considered invasive and is an agricultural pest in many regions outside its native range [START_REF] Barker | Molluscs as crop pests[END_REF][START_REF] Guiller | Tracing the invasion of the mediterranean land snail Cornu aspersum aspersum becoming an agricultural and garden pest in areas recently introduced[END_REF]).

Despite being one of the most studied land snail species (see references in Barker 2001b), there is no available information about its active dispersal in the literature. However, in C. aspersum and some related species, the level of locomotor activity has been shown to vary between lifehistory stages [START_REF] Oosterhoff | Variation in growth rate as an ecological factor in the landsnail Cepaea nemoralis (l.)[END_REF][START_REF] Fearnley | Sexual selection, dispersal and reproductive behaviour in hermaphrodite land snails, with particular reference to Helix aspersa M ller (Pulmonata: Gastropoda)[END_REF]. Even though dispersal was not measured in these studies, they suggest the existence of stage-dependent dispersal in helicid land snails. In this context, we studied how the first two phases of dispersal (i.e., departure and transience) differed between subadults and adults of C. aspersum, using individually marked snails in a closed semi-natural system. As snail activity is highly weather-dependent [START_REF] Bailey | The seasonal and daily patterns of locomotor activity in the snail Helix aspersa Müller, and their relation to environmental variables[END_REF], we included rainfall, temperature and humidity as covariates in our analyses. Because of the high energetic cost of movement in terrestrial molluscs [START_REF] Denny | Locomotion: the cost of gastropod crawling[END_REF], we expected adults to disperse more frequently and/or faster and further. Indeed, when dispersal is particularly costly, larger individuals are more likely to disperse, because they have the energy reserves to cope with these costs, and because they are more likely to successfully immigrate in a new, competitive patch (Bowler and Benton 2005).

Material and Methods

Collecting and maintenance of snails All individuals were collected at the end of the hibernation season (March 2011 andApril 2012) in wild populations from north-western France. Only individuals whose shell diameter was higher than 22 mm (i.e., stage II juveniles and adults, Klein-Rollais 1993) were used.

Individuals which had ended their somatic growth ("adults") were distinguished from stage II juveniles (hereafter named "subadults") by the presence of a reflected lip around the shell aperture (Klein-Rollais 1993). Subadults were on average smaller than adults (F 1,658 = 440.8, P < 0.001 ; mean shell diameter ± standard error in subadults: 26.5 ± 0.2 mm, n 2011+2012 = 329; in adults: 30.4 ± 0.1 mm, n 2011+2012 = 331). Snails were kept in controlled conditions (20 °C, 80 % R.H., 16L 8D photoperiod) for at least two weeks before release. They were housed in polythene boxes (30 × 45 × 8 cm) covered with a plastic mesh (30 individuals per box). To provide a moist substrate and a source of water for the snails, the bottom of each box was lined with synthetic foam kept saturated in water. Snails were fed ad libitum with a composite snail food (cereal powder supplemented with calcium; Ets Berton, France).

Monitoring of snail dispersal in semi-natural conditions

To study the effect of life-history stage and weather on dispersal movements, an experimental network was built in a meadow, on the Beaulieu university campus, Rennes (Brittany, France; 01°38'05'' W, 48°06'51'' N). This 13 x 50 m network included four 9 m² square patches of vegetation connected by four corridors, fenced with 15 cm high concrete block walls (Fig. 1).

To limit snail evasion and bird predation, the network was covered by mesh (mesh size: 18 mm in 2011, 12 mm in 2012). Two corridors were 1.2 m wide, while the others were 0.8 m wide. The ground was homogeneously covered by red fescue (Festuca rubra L.) and perennial ryegrass (Lolium perenne L.), and was gridded to ease the recording of snail coordinates. Wood and brick shelters were placed in each patch, big enough to shelter about 150 individuals per patch. Nettle (Urtica dioica L.) and ivy (Hedera helix L.) spots were planted (2 spots of each species per patch). Nettle leaves are particularly palatable to C. aspersum [START_REF] Iglesias | Field observations on feeding of the land snail Helix aspersa Müller[END_REF] and ivy is one of its main host plants in the wild [START_REF] Stratton | The non-marine Mollusca of the Parish of Dale[END_REF]. In 2011, drought impeded nettle and ivy growth, requiring the supplementation of food resources for this year; thus, 100 g of composite snail food per patch were added in plastic plates near the shelters.

Two climatic probes (Log32 datalogger, Dostmann, Reicholzheim, Germany) were installed at the western and eastern extremities of the network to record ground-level temperature and relative humidity at hourly intervals. As snails started to use northern and southern corridors in 2012, two supplementary probes were installed near the middle of these corridors (Fig. 1). Daily rainfall data were obtained from the Météo-France station of Rennes-St Jacques.

On 1st April 2011, 300 snails were released in the network. They were divided into four groups: two composed of 50 individuals (in the north-eastern and the south-western patches) and two composed of 100 individuals (in the north-western and the south-eastern patches). We initially chose different group sizes to determine if density affected dispersal. No significant effect of these density levels on dispersal metrics in 2011 was found (Generalized Linear Mixed Model with the emigration rate as a response variable and with weather and life-history stage as covariates, z = -0.223, P = 0.82). Thus, this distinction was abandoned for the 2012 session and the density effect not included in the complete analyses. After monitoring, the network was cleared of all remaining snails. On 4th June 2012, 360 new snails were released, divided into four equal groups of 90 individuals. To avoid an increase of dispersal due to translocation stress [START_REF] Turchin | Quantitative analysis of movement: measuring and modeling population redistribution in animals and plants[END_REF], release took place during the inactivity period (day), and snails were placed in or at the immediate proximity of the shelters. One hour after release, only a few individuals were outside the shelters (between 1 and 3, depending of the patch). Each released snail was marked with two numbered and coloured plastic marks commonly used for honeybees, glued with cyanoacrylate [START_REF] Henry | Marking hard-shelled gastropods: tag loss, impact on life-history traits, and perspectives in biology[END_REF]. Adults and subadults were in equal numbers in each population. Bonferroni-corrected Kolmogorov-Smirnov tests were used to ensure that shell size distributions of adults and subadults did not differ significantly between populations and between years. Individual locations were recorded twice a week for 52 days in 2011 and 25 days in 2012, with a week of interruption in 2012. Monitoring was prematurely interrupted in 2012 because of a high escape rate (see Results for a possible explanation): based on a Cormack-Jolly-Seber model with lost individuals considered "dead", only 36.1 % of the released individuals in 2012 were still in the network after 25 days in 2012 (130 snails of 360; 95 % confidence interval: 100 -168) against 73.0 % after 52 days in 2011 (219 snails of 300; 95 % CI: 208 -231). Snails were searched for in the experimental network itself and in a 50 cm wide buffer zone around it (Fig. 1). When a marked snail was found in the buffer zone, it was placed, after its coordinates were noted, at the nearest point inside the experimental network.

Statistical analyses

All statistical analyses were carried out using R (version 2.14; R Development Core Team 2011).

In this study, we considered as "resident" any snail found inside a circle of radius 1.20 m including its original shelter, nettle and ivy spots (Fig. 1). Indeed, C. aspersum snails do not usually forage that far when food is available near shelters [START_REF] Bailey | Daily cycles of feeding and locomotion in Helix aspersa[END_REF]). We assumed that recapture probability of residents was equal to 1. To avoid counting short explorations as dispersal, only snails that were not resident on at least two successive recapture occasions were considered "dispersers". We considered the same patch, observed two consecutive years, as two different patches in analyses.

To study the first step of dispersal, i.e., departure, we analysed the emigration probability between two successive recaptures as a function of life-history stage and weather using Generalized Linear Mixed Models (GLMMs) with the binomial error family and a logit link function. For each patch, emigration was analysed at the group level, as the proportion of individuals of a given life-history stage which dispersed between recaptures. Three weather variables with potential effect on snail activity were considered: mean daily rainfall, mean temperature and mean relative humidity. As the correlation between daily rainfall and mean relative humidity was weak (R² = 0.11), both variables were conserved in analyses. Because all recaptures were not separated by the same amount of time, we added an effect of interval duration. A year effect (factor with 2 levels) was also added. Finally, to test for differing responses to life-history stage and weather between years, interactions between these variables and year were added. The patch of origin was included as a random effect, to account for both spatial correlations (between subadults and adults from a same patch) and repeated measures.

To study the second step of dispersal, i.e., transience, we analysed two variables: the maximal dispersal distances observed and individual dispersal speeds. Maximal dispersal distances were analysed using linear models as a function of time since release and year, intercept being forced to zero. Dispersal speed is the speed at which an individual moves away from its original patch, calculated between the last recapture within the release patch and the last recapture as a disperser. Because, under the assumption of (correlated) random walk, the expected linear distance travelled does not increase linearly with time [START_REF] Turchin | Quantitative analysis of movement: measuring and modeling population redistribution in animals and plants[END_REF], and because dispersal duration differed between individuals, the ratio distance travelled/time is a biased measure of dispersal speed. To correct for this, we calculated for each individual its diffusion coefficient D = MSD / 4t (in m²/day), where MSD is the Mean Squared Displacement, i.e., the mean squared distance between recaptures, and t is the mean time between recaptures [START_REF] Turchin | Quantitative analysis of movement: measuring and modeling population redistribution in animals and plants[END_REF].

Excluding dispersers lost before they could be recaptured, this dataset contained 170 dispersing individuals (74 subadults, 96 adults). The distribution of diffusion coefficients was skewed, so we analysed the effect of weather and life-history stage using Gaussian GLMMs with a log link function. As for emigration, mean daily rainfall, mean temperature and mean relative humidity (at the nearest probe) were included as weather covariates. We considered both the patch of origin and the corridor used to disperse as random effects, to account for spatial correlations.

Here, no year effect was included, as most models with such an effect failed to converge. For both emigration rates and diffusion, a set of models was derived from the full models outlined above. These GLMMs were fitted using the lme4 R package (Bates et al. 2011). This gave a set of 275 possible models for emigration and 16 possible models for diffusion. Selection among these candidate models was realized in an information-theoretic framework, using the Akaike Information Criterion (AIC; [START_REF] Burnham | Model selection and multi-model inference: a practical information-theoretic approach[END_REF]. For each set, models were ranked by AIC, the model with the lowest AIC being considered the best and models with ΔAIC = AIC model -AIC best model ≤ 4 considered "good models". As each set contained more than one "good model", Akaike weights (summing to 1) were calculated and inferences were done by model averaging over the complete set of models [START_REF] Burnham | Model selection and multi-model inference: a practical information-theoretic approach[END_REF], using the MuMIn R package (Bartoń 2012). Averaging was done with shrinkage (predictors were averaged over all models, with their parameter values set to zero when they are absent from a given model).

Results

Weather differences between the two years (Table 1)

The 2012 session was significantly wetter than the 2011 session: wet days were more frequent in 2012 (χ² test, χ² = 4.84, df = 1, P = 0.03), mean daily rainfall was also higher (Mann-Whitney-Wilcoxon two-sample test, n 2011 = 52, n 2012 = 25, U = 363, P < 0.001), as well as relative humidity (ANOVA, F 1, 1868 = 254, P < 0.001). Mean temperature, however, was not significantly different between years (F 1, 1868 = 1.81, P = 0.18). The departure stage: emigration rate between recaptures at the patch level

The cumulated proportion of dispersers increased over time in both years (Fig. 2). In 2012, a proportion higher than 90 % was attained between 2 and 14 days after release and more than 30 % of snails emigrated as soon as the first day post release. In 2011, this 30 % proportion was not attained before 31 days post release. Material (ESM)). This proportion was independent of year and days since release (Fig. S1 in ESM), which would indicate the number of escapees depended only on the number of dispersers, the perimeter/surface ratio of the network, and the number of hidden openings.

Among the 275 models tested for the probability of dispersing between recaptures, 22 were considered "good" (ΔAIC ≤ 4; Table S1 in ESM). Thus, inferences on emigration were based on average coefficients, calculated over all models (Table 2). Among the three weather variables, the humidity effect was the only one different from zero, based on 95 % confidence intervals, and the only one to gather substantial support from data (evidence weight = 1, Table 2).

Emigration was on average positively correlated with mean humidity levels. There was an interaction between year and humidity: the increase of emigration in response to humidity was stronger in 2012 than in 2011 (Table 2, Fig. 3). Near saturation, the daily emigration rate was much higher than at intermediate levels of humidity (82.3 % of individuals leaving their patch per day of R.H. ≥ 90 %) (Fig. 3). Finally, emigration was stage-dependent, with subadults emigrating more than adults (evidence weight = 0.93, Table 2). Using the fact that, in GLMs and GLMMs with a logit link, the parameter value is the logarithm of the odds, we calculated that subadults were on average 1.16 times (exp(0.15)) more likely to emigrate than adults (95 % CI = [1.03; 1.31]).

The transience stage: individual diffusion coefficients during dispersal

Observed diffusion coefficients ranged between 0.06 and 28.92 m²/day in adults, and between 0.29 and 53.38 m²/day in subadults (mean adults ± SE = 5.84 ± 0.70 m²/day; mean subadults ± SE = 8.84 ± 1.21 m²/day). Among the 16 models analysed, four were considered "good" (ΔAIC ≤ 4;

Table S2 in ESM), including the full model. Thus, inferences on diffusion coefficients were based on average coefficients, calculated over all models (Table 3). Dispersal speed was on average positively correlated with all three weather variables: mean rainfall, mean relative humidity and mean temperature between successive recaptures (Table 3). However, the mean rainfall effect was the only one different from zero, based on 95 % confidence intervals (Table 3). Furthermore, the mean rainfall effect was the only one to gather substantial support from data, with an evidence weight of 0.93 (evidence weights are 0.46 and 0.37 for the relative humidity and the temperature effect, respectively). Independently of weather conditions, subadults were on average faster than adults, with an evidence weight of 0.91 for this effect of life-history stage (Table 3, Fig. 5). 3), based on fixed effects only, are plotted (solid lines) with their 95 % confidence intervals (broken lines).

Discussion

Compared to adults, subadults of the brown garden snail Cornu aspersum emigrated more and moved faster and further during transience. Furthermore, we also found that humidity levels and rainfall were the main weather variables affecting both emigration and transience. If the effects of life-history stage and weather (temperature and humidity) on the initiation of dispersal are relatively well-studied (e.g., [START_REF] Tuda | Relative importance of weather and density dependence on the dispersal and on-plant activity of the predator Orius minutus[END_REF][START_REF] Bowler | Variation in dispersal mortality and dispersal propensity among individuals: the effects of age, sex and resource availability[END_REF], their effects during the transience phase have been much less investigated (with the exception of wind in flying insects and passively dispersing organisms; e.g., [START_REF] Reynolds | Ballooning dispersal in arthropod taxa: conditions at take-off[END_REF]). Here, overall effects of life-history stage and weather were similar between emigration and transience. This would indicate the existence of an integrated dispersal strategy, maximizing general dispersal ability (both emigration propensity and dispersal speed) at the life-history stage where benefits of dispersal are maximal (and/or costs minimal).

Weather conditions have a major influence on ectotherms' metabolism and activity; thus, they are usually expected to influence dispersal in similar ways [START_REF] Delattre | Influence de la structure du paysage et des conditions météorologiques sur le comportement de dispersion de Maniola jurtina (Lepidoptera : Nymphalidae, L.) dans un agroécosystème bocager[END_REF]. Here, wet weather increased both emigration probability and dispersal speed of land snails. Moreover, in 2012, maximal dispersal distances were close to what is observed in other snails and slugs of similar size [START_REF] Aubry | Active and passive dispersal of an invading land snail in Mediterranean France[END_REF][START_REF] Honek | Body size and the colonisation of cereal crops by the invasive slug Arion lusitanicus[END_REF], but they were much lower in 2011, which was a dry year. Because land snails and slugs can experience significant water loss through their integument and through mucus secretion, they generally slow down their activity, or even enter a state of dormancy under dry conditions [START_REF] Cook | Behavioural ecology: on doing the right thing, in the right place at the right time[END_REF]. The present study logically extends this well-documented weather-dependence of general activity to dispersal movements.

However, the average model for the emigration rate also contains a strong humidity × year interaction, with an effect of humidity much stronger in 2012 (Table 2). As humidity ranges in both years did not overlap much (Fig. 3), it is difficult to tell whether this reflects a real difference between years or a non-linear effect of humidity, independent of the year. Prior to release, snails were fed with a composed food. In 2011 only, this resource was added in patches, to compensate for the drought-induced low growth of host plants. Cornu aspersum can acquire a strong preference toward a given food resource after conditioning (e.g., [START_REF] Peschel | Consequences of food-attraction conditioning in Helix: a behavioral and electrophysiological study[END_REF]. It is possible that the absence of this familiar food in 2012 rendered the patches less attractive, encouraging emigration. During the 2012 session, wetter conditions also led to an increase in vegetation height and cover in corridors. As a consequence, the contrast between patches and corridors was less marked, compared to 2011. The perception of such contrasts is one of the main proximate mechanisms limiting movement of animals within patches of favourable resources [START_REF] Haynes | Interpatch movement and edge effects: the role of behavioral responses to the landscape matrix[END_REF]. When there is no clear boundary between patches and matrix (as in 2012), dispersal can occur as a succession of routine movements [START_REF] Baguette | Landscape connectivity and animal behavior: functional grain as a key determinant for dispersal[END_REF]. Alternatively, there are indications that high humidity levels have deleterious effects in C. aspersum [START_REF] Klein-Rollais | Contribution à l'étude de la balance hydrique et de sa régulation chez l'escargot Petit-Gris[END_REF]. When maintained at humidity levels above 90 %, snails become unable to regulate correctly their water balance and hyperhydrate. Daily activity rhythms are disturbed, some individuals lose their shell after their columellar muscle break, and mortality increases to up to 20 times the level observed in optimal conditions (100 % against 4.7 % in 2 months, Klein-Rollais 1993). Thus, it is possible that the observed non-linear response to humidity was independent of the year; in this context, the very high emigration rates observed at high humidity levels could be an emergency response to potentially lethal conditions. Nonlinear weather effects and differences in patch quality are both known to influence dispersal (e.g., Delattre 2010 for the former; [START_REF] Mathieu | Habitat quality, conspecific density, and habitat pre-use affect the dispersal behaviour of two earthworm species, Aporrectodea icterica and Dendrobaena veneta, in a mesocosm experiment[END_REF] for the latter). Experiments in strongly contrasted natural or semi-natural landscapes, covering more completely the range of weather conditions experienced by C. aspersum, would help to correctly disentangle the respective effects of these environmental conditions in its dispersal.

The existence of a specific dispersing life-history stage, and its nature, depend on the selective pressures on dispersal and/or movement constraints, and how they vary with age or size [START_REF] Bowler | Variation in dispersal mortality and dispersal propensity among individuals: the effects of age, sex and resource availability[END_REF]Benton 2005, 2009). Dispersal is a risky behaviour, entailing many costs (Bonte et al.

2012

). When costs are high, larger individuals are expected to be better dispersers (Benard and McCauley 2008). In several mammal species for instance, there is a threshold body mass below which individuals do not disperse (e.g., [START_REF] Debeffe | Condition-dependent natal dispersal in a large herbivore: heavier animals show a greater propensity to disperse and travel further[END_REF]. Based on this and the high energetic cost of movement in land snails [START_REF] Denny | Locomotion: the cost of gastropod crawling[END_REF], we hypothesized that adults, which are larger, would be more likely to emigrate and would be faster dispersers than subadults. Indeed, in the invasive slug Arion lusitanicus, larger individuals emigrate more and are the fastest dispersers [START_REF] Honek | Body size and the colonisation of cereal crops by the invasive slug Arion lusitanicus[END_REF]. Here, the relationship between life-history stage and dispersal in Cornu aspersum shows the opposite pattern: subadults are more likely to emigrate and are faster than adults during transience. This is especially surprising for transience: because they have a thinner shell, subadults of C. aspersum tend to lose water faster than adults (Klein-Rollais 1993), which would make them more vulnerable to desiccation during prolonged movement. This suggests direct costs of locomotion are not the main driving force acting on dispersal. Indeed, the fastest observed snails dispersed 10 m in 2 days or less, and 45 m in up to 23 days, which is only about half the speed of some walking insects (e.g., [START_REF] Schooley | Movements of cactus bugs: Patch transfers, matrix resistance, and edge permeability[END_REF], when energetic costs of movement are theoretically much higher [START_REF] Denny | Locomotion: the cost of gastropod crawling[END_REF].

Altogether, this hints at the existence of other, non-energy related, stage-or size-dependent benefits and costs of dispersal.

Negative size-dependent dispersal, where smaller individuals disperse more, has been associated with intraspecific competition (Bowler and Benton 2005). When the cost of competition is higher than the cost of dispersal, inferior competitors are expected to be more prone to emigrate.

This could be the case in Cornu aspersum, as subadults do suffer from interference competition when placed with adults [START_REF] Dan | Growth, mortality, and feeding rates of the snail Helix aspersa at different population densities in the laboratory, and the depression of activity of helicid snails by other individuals, or their mucus[END_REF]. Intraspecific competition can also affect dispersal distances. When the cost of moving further is lower than the cost of settling in, or close to, a highly competitive patch, actively dispersing animals should pass over this patch and continue to disperse, as recently demonstrated in the two-spotted mite (Bitume et al. 2013). In this context, less competitive individuals would move further, as observed in our setup.

However, we did not observe an effect of density on dispersal in 2011 (see Material and Methods), which would be expected if dispersal was driven by intraspecific competition. It should be noted that the initial densities we used were in the higher range of natural population densities, and closer to those observed in extensive snail farms [START_REF] Dan | Studies on the growth and ecology of Helix aspersa Müller[END_REF][START_REF] González | Effect of population density on snail productivity (Helix aspersa) in an open sky system fed with Swiss chard and a balanced food supplement[END_REF]. Using densities closer to those observed in natural populations should help to evaluate this possible effect of competition on dispersal rates and dispersal distances.

Dispersal prior to sexual maturation can also be selected as a means to avoid inbreeding, especially when individuals are unable to recognize kin (Bowler and Benton 2005). Helicid snails seem unable to discriminate between siblings and non-siblings when it comes to mate choices (Baur and Baur 1997) or egg cannibalism (Baur 1990). In this context, avoiding matings between relatives is likely to be one of the main drivers of subadult dispersal in C. aspersum, which is moreover a species extremely sensitive to the deleterious effects of inbreeding (Albuquerque de Matos and Serra 1984). Thus, subadult dispersal in the brown garden snail could be seen, as in many other groups, as a case of natal dispersal, i.e., the movement from the site of birth to the site of first reproduction [START_REF] Clobert | Dispersal ecology and evolution[END_REF]). However, interpreting the subadult dispersal patterns observed here as natal dispersal stricto sensu is difficult for at least two reasons. First, pre-capture dispersal and reproductive histories of the individuals tested here are unknown. Then, in helicid land snails, subadults are able to mate, produce sperm and lay eggs, albeit with less success than adults [START_REF] Fearnley | Sexual selection, dispersal and reproductive behaviour in hermaphrodite land snails, with particular reference to Helix aspersa M ller (Pulmonata: Gastropoda)[END_REF][START_REF] Kupfernagel | Sperm utilization in subadult and adult simultaneous hermaphrodite snails mating in the wild[END_REF] [START_REF] Baeza | Male mating opportunities affect sex allocation in a protandric-simultaneous hermaphroditic shrimp[END_REF]. Male subadults can mate with many snails without having to pay the cost of egg creation, when adults must pay these costs, and thus have less energy to invest in movement. In these conditions, according to [START_REF] Tomiyama | Dispersal patterns of the giant African snail, Achatina fulica (Férussac) (Stylommatophora: Achatinidae), equipped with a radio-transmitter[END_REF], natural selection should favour high dispersal and multiple matings in male subadults, and sedentarity in fully hermaphroditic adults. This hypothesis focuses on the energetic costs of reproduction, but such a dispersal strategy would also lead to reduced inbreeding by scattering related snails before their full sexual maturity. This has yet to be fully tested, but there is indication that C. aspersum might be a protandric simultaneous hermaphroditic species, with subadults being functionally male [START_REF] Bride | Asynchronisme du développement du tractus génital de l'escargot Helix aspersa pendant la croissance et la reproduction[END_REF][START_REF] Fearnley | Sexual selection, dispersal and reproductive behaviour in hermaphrodite land snails, with particular reference to Helix aspersa M ller (Pulmonata: Gastropoda)[END_REF].

Dispersal is a complex process, under multiple, simultaneous and sometimes contradictory selective pressures [START_REF] Clobert | Dispersal ecology and evolution[END_REF]). Thus, it is possible, and even likely, that the three mechanisms proposed here (namely competition avoidance, natal dispersal linked to inbreeding avoidance and differences in costs between "males" and hermaphrodites if there is protandric simultaneous hermaphroditism), simultaneously influence, to varying degrees, C. aspersum dispersal. Their relative importance could have profound impacts on metapopulation dynamics and the persistence of local populations. For instance, if subadult dispersal is mainly linked to mate search, as in [START_REF] Tomiyama | Dispersal patterns of the giant African snail, Achatina fulica (Férussac) (Stylommatophora: Achatinidae), equipped with a radio-transmitter[END_REF] hypothesis, immigration should be higher in high-density patches. This would increase the extinction risk of small populations, as dispersers settle in already large populations and empty patches are not recolonized [START_REF] Saether | Finite metapopulation models with density-dependent migration and stochastic local dynamics[END_REF]).

On the other hand, when dispersal is mainly a response to competition, dispersers tend to settle more in low-density patches, increasing the local growth rate and decreasing the local extinction risk (demographic rescue effect; [START_REF] Saether | Finite metapopulation models with density-dependent migration and stochastic local dynamics[END_REF][START_REF] Hanski | Dispersal and eco-evolutionary dynamics in the Glanville fritillary butterfly[END_REF]. Moreover, dispersal before reproductive maturity and the arrival of unrelated individuals would decrease inbreeding depression and its deleterious effects, especially in low-density patches (genetic rescue effect;

Figure 1

 1 Figure 1 Schematic representation of a patch in the experimental network (dark grey), showing the disposition of corridors and the buffer zone (light grey) where any individual found was put back in the network. The resources available to snails inside patches are also represented: rectangles = shelters; triangles = nettle spots; stars = ivy spots; circles = composite snail food in plastic plates (in 2011 only). The dotted circle marks the limit beyond which snails were considered dispersers. Inset: a representation of the entire network. The asterisks mark the location of the climatic probes.

Figure 2 :

 2 Figure 2: Cumulated proportion of dispersing snails as a function of time since release and year. For each year, data from the 4 patches are pooled; n 2011 = 300, n 2012 = 360. During the two monitoring sessions, 423 snails (90 in 2011, 333 in 2012) moved more than

  Parameter weight: the sum of the AIC weights of all models in which the parameter appears. Subadult: effect of the life-history stage being "subadult", compared to adults; Interval duration: time between successive recaptures; R.H.: mean relative humidity; Temp.: mean temperature; Rain: mean daily rainfall. Interactions et een the year effe t and other aria les are denoted y a × sign. *: average parameter value different from 0, based on 95 % confidence intervals.

Figure 3 :

 3 Figure3: Mean (± 95 % confidence intervals) daily emigration rates observed for five 10 % wide classes of relative humidity, depending of the year. When intervals between recaptures were longer than one day, daily emigration rates were estimated as 1 -(proportion of nondispersing snails between recaptures) 1 / days between recaptures . 95 % confidence intervals are ± 1.96 SE.

Figure 4 :

 4 Figure 4: Maximal dispersal distances observed within the experimental network, as a function of year and time since release. Predicted values (lines) are based on a linear model with days as covariate and year as a factor (n = 23, R² = 0.97). y 2011 = 0.268 x, y 2012 = 2.576 x, the intercept values being set to zero by construction.

Figure 5 :

 5 Figure 5: Observed (circles) and predicted (lines) mean diffusion coefficients of dispersing subadults (n = 74, left) and adults (n = 90, right) of Cornu aspersum, as a function of mean daily rainfall during dispersal.For ease of reading, each circle represent the mean observed diffusion coefficient of a group (individuals from the same patch, dispersing via the same corridor, see Materials and Methods), for a given mean rainfall level (rounded to the nearest mm). The size of each circle is proportional to the number of individuals. Predictions from the best GLMM (Table3), based on fixed effects only, are plotted (solid lines) with their 95 % confidence intervals (broken lines).

  [START_REF] Saccheri | Rapid spread of immigrant genomes into inbred populations[END_REF][START_REF] Hanski | Dispersal and eco-evolutionary dynamics in the Glanville fritillary butterfly[END_REF]. In order to better understand the relative importance of these mechanisms, the effect of inter-individual variation in size, age and reproductive development on all steps of dispersal (including immigration) should be investigated. Overall, our results highlight the necessity of studying more diverse reproductive systems regarding dispersal, as a similar pattern might be caused by different underlying strategies in hermaphrodites than in gonochoric animals. Such "alternative" dispersal strategies could have many interesting, yet unstudied, consequences in terms of life-history evolution, sexual selection and population dynamics. Bates D, Maechler M, Bolker B (2011) R package lme4: linear mixed-effects models using S4 classes. http://lme4.r-forge.r-project.org/ Baur B (1990) Egg cannibalism in hatchlings of the land snail Helix pomatia: nutritional advantage may outweigh lack of kin recognition. Malacol Rev 23:103-105 Baur A (1993a) Effects of food availability and intraspecific and interspecific interactions on the dispersal tendency in the land snail Chondrina clienta. J Zool 230:87-100 Baur B (1993b) Population structure, density, dispersal and neighbourhood size in Arianta arbustorum. Ann Naturhist Mus Wien 94/95:307-321 Baur B, Baur A (1997) Random mating with respect to relatedness in the simultaneously hermaphroditic land snail Arianta arbustorum. Invertebr Biol 116:294 Benard MF, McCauley SJ (2008) Integrating across life-history stages: consequences of natal habitat effects on dispersal. Am Nat 171:553-567 Bitume EV, Bonte D, Ronce O, Bach F, Flaven E, Olivieri I, Nieberding CM (2013) Density and genetic relatedness increase dispersal distance in a subsocial organism. Ecol Lett 16:430-437 Bonte D, Van Dyck H, Bullock JM, Coulon A, Delgado M, Gibbs M, Lehouck V, Matthysen E, Mustin K, Saastamoinen M, Schtickzelle N, Stevens VM, Vandewoestijne S, Baguette M, Barton K, Benton TG, Chaput-Bardy A, Clobert J, Dytham C, Hovestadt T, Meier CM, Palmer SCF, Turlure C, Travis JMJ (2012) Costs of dispersal. Biol Rev 87:290-312 Bowler DE, Benton TG (2005) Causes and consequences of animal dispersal strategies: relating individual behaviour to spatial dynamics. Biol Rev 80:205-225

Table 1 :

 1 Weather differences between the two field sessions(2011 and 2012).Mean values are presented ± SE. Mean values for temperature and humidity are those recorded at the eastern probe. Values from other probes are not significantly different (P > 0.05 for all variables). ***: P < 0.001; see text for details of tests.

	2011	2012	P

Table 2 :

 2 Averaged model parameters for the emigration probability between successive recaptures. Parameter values are averaged over the complete set of models with shrinkage (see TableS1in ESM).

		Average effect ± SE	Parameter weight
	Subadult	0.15 ± 0.06 *	0.93
	Interval duration	0.04 ± 0.01 *	1.00
	R.H.	0.20 ± 0.03 *	1.00
	Year	-22.93 ± 7.51 *	1.00
	Year × R.H.	0.31 ± 0.09 *	1.00
	(Intercept)	-16.27 ± 3.23 *	1.00
	Temp.	-0.07 ± 0.08	0.43
	Rain	-0.02 ± 0.09	0.71
	Year × Subadult	0.02 ± 0.14	0.23
	Year × Interval duration	-0.02 ± 0.02	0.34
	Year × Temp.	0.11 ± 0.29	0.12
	Year × Rain	-0.23 ± 0.13	0.44

Table 3 :

 3 Averaged model parameters for the individual diffusion coefficients of dispersingCornu aspersum snails. Parameter values are averaged over the complete set of models with shrinkage (see TableS2in ESM).

	Average effect ± SE	Parameter weight
	Subadult	0.42 ± 0.08 *	0.91
	Rain	0.27 ± 0.06 *	0.92
	Temp.	0.05 ± 0.07	0.37
	R.H.	0.02 ± 0.02	0.46
	(Intercept)	0.92 ± 2.50	1.00
	Parameter weight: the sum of the AIC weights of all models in which the parameter appears. Subadult:
	effect of the life-history stage being "subadult", compared to adults; Rain: mean daily rainfall; Temp.:
	mean temperature; R.H.: mean relative humidity. *: average parameter value different from 0, based on
	95 % confidence intervals.

  , making it difficult to equal subadult/adult with pre-breeding/post-breeding. Laboratory experiments, where the age and the reproductive history of individuals can be known easily, are needed to determine whether subadult dispersal in C. aspersum really corresponds to natal dispersal.[START_REF] Tomiyama | Dispersal patterns of the giant African snail, Achatina fulica (Férussac) (Stylommatophora: Achatinidae), equipped with a radio-transmitter[END_REF] proposed an alternative hypothesis to explain similar dispersal patterns in the giant snail Achatina fulica. Achatina fulica is one of the few animal species known to present a protandric simultaneous hermaphroditism, i.e., a mixed sex allocation pattern in which individuals mature and reproduce initially as males, and later become functional simultaneous hermaphrodites

Acknowledgements

We thank Eric Petit and four anonymous referees for their very helpful comments on previous versions of the manuscript, which greatly improved the quality of this article. We are grateful to Thierry Fontaine and Fouad Nassur for their precious help in building and (in 2012) repairing the experimental network. We also thank the University of Rennes 1 for allowing us to use part of the Evenière park and for technical support.

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