Keywords: 1991 Mathematics Subject Classification. 18B40, 20N02, 20F36 Garside category, Garside monoid, self-distributivity, braid, greedy normal form, least common multiple, LD-expansion

is more relevant for that notion, which involves no counterpart of any Garside element or map, but is only the common substratum of all Garside structures.

2008-10

LEFT-GARSIDE CATEGORIES, SELF-DISTRIBUTIVITY, AND BRAIDS

PATRICK DEHORNOY

Abstract. In connection with the emerging theory of Garside categories, we develop the notions of a left-Garside category and of a locally left-Garside monoid. In this framework, the connection between the self-distributivity law LD and braids amounts to the result that a certain category associated with LD is a left-Garside category, which projects onto the standard Garside category of braids. This approach leads to a realistic program for establishing the Embedding Conjecture of [Dehornoy, Braids and Self-distributivity, Birkhaüser (2000), Chap. IX].

The notion of a Garside monoid emerged at the end of the 1990's [START_REF] Dehornoy | Gaussian groups and Garside groups, two generalizations of Artin groups[END_REF][START_REF] Dehornoy | Groupes de Garside[END_REF] as a development of Garside's theory of braids [START_REF] Garside | The braid group and other groups[END_REF], and it led to many developments [2, 3, 5, 6, 7, 8, 13, 14, 15, 31, 33, 34, 41, 42, 45, 46, 47, ...]. More recently, Bessis [START_REF] Bessis | Garside categories, periodic loops and cyclic sets[END_REF], Digne-Michel [START_REF] Digne | Garside and locally Garside categories[END_REF], and Krammer [START_REF] Krammer | A class of Garside groupoid structures on the pure braid group[END_REF] introduced the notion of a Garside category as a further extension, and they used it to capture new, nontrivial examples and improve our understanding of their algebraic structure. The concept of a Garside category is also implicit in [START_REF] Deligne | Representations of reductive groups over finite fields[END_REF] and [START_REF] Godelle | Normalisateurs et centralisateurs des sous-groupes paraboliques dans les groupes d'Artin-Tits[END_REF], and maybe in the many diagrams of [START_REF] Dehornoy | Braids and Self-Distributivity[END_REF].

Here we shall describe and investigate a new example of (left)-Garside category, namely a certain category LD + associated with the left self-distributivity law (LD)

x(yz) = (xy)(xz).

The interest in this law originated in the discovery of several nontrivial structures that obey it, in particular in set theory [START_REF] Dehornoy | 1 1 -complete families of elementary sequences[END_REF][START_REF] Laver | The left distributive law and the freeness of an algebra of elementary embeddings[END_REF] and in low-dimensional topology [START_REF] Joyce | A classifying invariant of knots: the knot quandle[END_REF][START_REF] Fenn | Racks and links in codimension 2[END_REF][START_REF] Matveev | Distributive groupoids in knot theory[END_REF]. A rather extensive theory was developed in the decade 1985-95 [START_REF] Dehornoy | Braids and Self-Distributivity[END_REF]. Investigating self-distributivity in the light of Garside categories seems to be a good idea. It turns out that a large part of the theory developed so far can be summarized into one single statement, namely

The category LD + is a left-Garside category, stated as the first part of Theorem 6.1.

The interest of the approach should be at least triple. First, it gives an opportunity to restate a number of previously unrelated properties in a new language that is natural and should make them more easily understandable-this is probably not useless. In particular, the connection between self-distributivity and braids is now expressed in the simple statement:

There exists a right-lcm preserving surjective functor of LD + to the Garside category of positive braids, (second part of Theorem 6.1). This result allows one to recover most of the usual algebraic properties of braids as a direct application of the properties of LD + : roughly speaking, Garside's theory of braids is the emerged part if an iceberg, namely the algebraic theory of self-distributivity. Second, a direct outcome of the current approach is a realistic program for establishing the Embedding Conjecture. The latter is the most puzzling open question involving free self-distributive systems. Among others, it says that the equivalence class of any bracketed expression under self-distributivity is a semilattice, i.e., any two expressions admit a least upper bound with respect to a certain partial ordering. Many equivalent forms of the conjecture are know [START_REF] Dehornoy | Braids and Self-Distributivity[END_REF]Chapter IX]. At the moment, no complete proof is known, but we establish the following new result Unless the left-Garside category LD + is not regular, the Embedding Conjecture is true, (Theorem 6.2). This result reduces a possible proof of the conjecture to a (long) sequence of verifications.

Third, the category LD + seems to be a seminal example of a left-Garside category, quite different from all previously known examples of Garside categories. In particular, being strongly asymmetric, LD + is not a Garside category. The interest of investigating such objects per se is not obvious, but the existence of a nontrivial example such as LD + seems to be a good reason, and a help for orientating further reaserch. In particular, our approach emphasizes the role of locally left-Garside monoids 1 : this is a monoid M that fails to be Garside because no global element ∆ exists, but nevertheless possesses a family of elements ∆ x that locally play the role of the Garside element and are indexed by a set on which the monoid M partially acts. Most of the properties of left-Garside monoids extend to locally left-Garside monoids, in particular the existence of least common multiples and, in good cases, of the greedy normal form.

Acknowledgement. Our definition of a left-Garside category is borrowed from [START_REF] Digne | Garside and locally Garside categories[END_REF] (up to a slight change in the formal setting, see Remark 2.6). Several proofs in Section 2 and 3 use arguments that are already present, in one form or another, in [START_REF] Adyan | Fragments of the word Delta in a braid group[END_REF][START_REF] Thurston | Finite state algorithms for the braid group[END_REF][START_REF] Elrifai | Algorithms for positive braids[END_REF][START_REF] Epstein | Word Processing in Groups[END_REF][START_REF] Charney | Artin groups of finite type are biautomatic[END_REF][START_REF] Dehornoy | Groupes de Garside[END_REF][START_REF] Godelle | Normalisateurs et centralisateurs des sous-groupes paraboliques dans les groupes d'Artin-Tits[END_REF] and now belong to folklore. Most appear in the unpublished paper [START_REF] Digne | Garside and locally Garside categories[END_REF] by Digne and Michel, and are implicit in Krammer's paper [START_REF] Krammer | A class of Garside groupoid structures on the pure braid group[END_REF]. Our reasons for including such arguments here is that adapting them to the current weak context requires some polishing, and that it makes it natural to introduce our two main new notions, namely locally Garside monoids and regular left-Garside categories.

The paper is organized in two parts. The first one (Sections 1 to 3) contains those general results about left-Garside categories and locally left-Garside monoids that will be needed in the sequel, in particular the construction and properties of the greedy normal form. The second part (Sections 4 to 8) deals with the specific case of the category LD + and its connection with braids. Sections 4 and 5 review basic facts about the self-distributivity law and explain the construction of the category LD + . Section 6 is devoted to proving that LD + is a left-Garside category and to showing how the results of Section 3 might lead to a proof of the Embedding Conjecture. In Section 7, we show how to recover the classical algebraic properties of braids from those of LD + . Finally, we explain in Section 8 some alternative solutions for projecting LD + to braids. In an appendix, we briefly describe what happens when the associativity law replaces the self-distributivity law: here also a left-Garside category appears, but a trivial one.

We use N for the set of all positive integers.

Left-Garside categories

We define left-Garside categories and describe a uniform way of constructing such categories from so-called locally left-Garside monoids, which are monoids with a convenient partial action.

1.1. Left-Garside monoids. Let us start from the now classical notion of a Garside monoid. Essentially, a Garside monoid is a monoid in which divisibility has good properties, and, in addition, there exists a distinguished element ∆ whose divisors encode the whole structure. Slightly different versions have been considered [START_REF] Dehornoy | Gaussian groups and Garside groups, two generalizations of Artin groups[END_REF][START_REF] Dehornoy | Groupes de Garside[END_REF][START_REF] Digne | Présentations duales pour les groupes de tresses de type affine Atilde[END_REF], the one stated below now being the most frequently used. In this paper, we are interested in one-sided versions involving left-divisibility only, hence we shall first introduce the notion of a left-Garside monoid.

Throughout the paper, if a, b are elements of a monoid-or, from Section 1.2, morphisms of a category-we say that a left-divides b, denoted a b, if there exists c satisfying ac = b. The set of all left-divisors of a is denoted by Div(a). If ac = b holds with c = 1, we say that a is a proper left-divisor of b, denoted a ≺ b.

We shall always consider monoids M where 1 is the only invertible element, which will imply that is a partial ordering. If two elements a, b of M admit a greatest lower bound c with respect to , the latter is called a greatest common left-divisor, or left-gcd, of a and b, denoted c = gcd(a, b). Similarly, a -least upper bound d is called a least common right-multiple, or right-lcm, of a and b, denoted d = lcm(a, b). We say that M admits local right-lcm's if any two elements of M that admit a common right-multiple admit a right-lcm.

Finally, if M is a monoid and S, S ′ are subsets of M , we say that S leftgenerates S ′ if every nontrivial element of S ′ admits at least one nontrivial leftdivisor belonging to S. Definition 1.1. We say that a monoid M is left-preGarside if (LG 0 ) for each a in M , every ≺-increasing sequence in Div(a) is finite, (LG 1 ) M is left-cancellative, (LG 2 ) M admits local right-lcm's.

An element ∆ of M is called a left-Garside element if (LG 3 ) Div(∆) left-generates M , and a ∆ implies ∆ a∆.

We say that M is left-Garside if it is left-preGarside and possesses at least one left-Garside element.

Using "generates" instead of "left-generates" in (LG 3 ) would make no difference, by the following trivial remark-but the assumption (LG 0 ) is crucial, of course. Lemma 1.2. Assume that M is a monoid satisfying (LG 0 ). Then every subset S left-generating M generates M .

Proof. Let a be a nontrivial element of M . By definition there exist a 1 = 1 in S and a ′ satisfying a = a 1 a ′ . If a ′ is trivial, we are done. Otherwise, there exist a 2 = 1 in S and a ′′ satisfying a ′ = a 2 a ′′ , and so on. The sequence 1, a 1 , a 1 a 2 , ... is ≺increasing and it lies in Div(a), hence it must be finite, yielding a = a 1 ...a d with a 1 , ..., a d in S. The equivalence of the above definition with that of [START_REF] Digne | Présentations duales pour les groupes de tresses de type affine Atilde[END_REF] is easily checked. The seminal example of a Garside monoid is the braid monoid B + n equipped with Garside's fundamental braid ∆ n , see for instance [START_REF] Garside | The braid group and other groups[END_REF][START_REF] Epstein | Word Processing in Groups[END_REF]. Other classical examples are free abelian monoids and, more generally, all spherical Artin-Tits monoids [START_REF] Brieskorn | Artin-Gruppen und Coxeter-Gruppen[END_REF], as well as the so-called dual Artin-Tits monoids [START_REF] Birman | A new approach to the word problem in the braid groups[END_REF][START_REF] Bessis | Garside categories, periodic loops and cyclic sets[END_REF]. Every Garside monoid embeds in a group of fractions, which is then called a Garside group.

Right

Let us mention that, if a monoid M is left-Garside, then mild conditions imply that it is Garside: essentially, it is sufficient that M is right-cancellative and that the left-and right-divisors of the left-Garside element ∆ coincide [START_REF] Dehornoy | Groupes de Garside[END_REF].

1.2. Left-Garside categories. Recently, it appeared that a number of results involving Garside monoids still make sense in a wider context where categories replace monoids [START_REF] Bessis | Garside categories, periodic loops and cyclic sets[END_REF][START_REF] Digne | Garside and locally Garside categories[END_REF][START_REF] Krammer | A class of Garside groupoid structures on the pure braid group[END_REF]. A category is similar to a monoid, but the product of two elements is defined only when the target of the first is the source of the second. In the case of Garside monoids, the main benefit of considering categories is that it allows for relaxing the existence of the global Garside element ∆ into a weaker, local version depending on the objects of the category, namely a map from the objects to the morphisms.

We refer to [START_REF] Lane | Categories for the Working Mathematician[END_REF] for some basic vocabulary about categories-we use very little of it here.

Convention. Throughout the paper, composition of morphisms is denoted by a multiplication on the right: f g means "f then g". If f is a morphism, the source of f is denoted ∂ 0 f , and its target is denoted ∂ 1 f . In all examples, we shall make the source and target explicit: morphisms are triples (x, f, y) satisfying

∂ 0 (x, f, y) = x, ∂ 1 (x, f, y) = y.
A morphism f is said to be nontrivial if f = 1 ∂0f holds.

We extend to categories the terminology of divisibility. So, we say that a morphism f is a left-divisor of a morphism g, denoted f g, if there exists h satisfying f h = g. If, in addition, h can be assumed to be nontrivial, we say that f ≺ g holds. Note that f g implies ∂ 0 f = ∂ 0 g. We denote by Div(f ) the collection of all left-divisors of f .

The following definition is equivalent to Definition 2.10 of [START_REF] Digne | Garside and locally Garside categories[END_REF] by F. Digne and J. Michel-see Remark 2.6 below. Definition 1.4. We say that a category C is left-preGarside if (LG 0 ) for each f in Hom(C), every ≺-ascending sequence in Div(f ) is finite, (LG 1 ) Hom(C) admits left-cancellation, (LG 2 ) Hom(C) admits local right-lcm's. A map ∆ : Obj(C) → Hom(C) is called a left-Garside map if, for each object x, we have ∂ 0 ∆(x) = x and (LG 3 ) ∆(x) left-generates Hom(x, -), and f ∆(x) implies ∆(x) f ∆(∂ 1 f ). We say that C is left-Garside if it is left-preGarside and possesses at least one left-Garside map.

Example 1.5. Assume that M is a left-Garside monoid with left-Garside element ∆. One trivially obtains a left-Garside category C(M ) by putting

Obj(C(M )) = {1}, Hom(C(M )) = {1} × M × {1}, ∆(1) = ∆.
Another left-Garside category C M can be attached with M , namely taking

Obj( C(M )) = M, Hom( C(M )) = {(a, b, c) | ab = c }, ∆(a) = ∆.
It is natural to call C(M ) the Cayley category of M since its graph is the Cayley graph of M (defined provided M is right-cancellative).

The notion of a right-Garside category can be defined symmetrically, exchanging left and right everywhere and exchanging the roles of source and target. In particular, the map ∆ and Axiom (LG 3 ) is to be replaced by a map ∇ satisfying ∂ 1 ∇(x) = x and, using b a for "a right-divides b", (LG 3 ) ∇(y) right-generates Hom(-, y), and ∇(y) f implies ∇(∂ 0 y)f ∇(y). Then comes the natural two-sided version of a Garside category [START_REF] Bessis | Garside categories, periodic loops and cyclic sets[END_REF][START_REF] Digne | Garside and locally Garside categories[END_REF]. Definition 1.6. We say that a category C is Garside with Garside map ∆ if C is left-Garside with left-Garside map ∆ and right-Garside with right-Garside map ∇ satisfying ∆(x) = ∇(∂ 1 (∆(x)) and ∇(y) = ∆(∂ 0 (∇(y)) for all objects x, y.

It is easily seen that, if M is a Garside monoid, then the categories C(M ) and C(M ) of Example (1.5) are Garside categories. Insisting that the maps ∆ and ∇ involved in the left-and right-Garside structures are connected as in Definition 1.6 is crucial: see Appendix for a trivial example where the connection fails.

1.3.

Locally left-Garside monoids. We now describe a general method for constructing a left-Garside category starting from a monoid equipped with a partial action on a set. The trivial examples of Example 1.5 enter this family, and so do the two categories LD + and B + investigated in the second part of this paper. Definition 1.7. Assume that M is a monoid. We say that α :

M × X → X is a partial (right) action of M on X if, writing x • a for α(a)(x), (i) x • 1 = x holds for each x in X, (ii) (x • a) • b = x • ab
holds for all x, a, b, this meaning that either both terms are defined and they are equal, or neither is defined, (iii) for each finite subset S in M , there exists x in X such that x • a is defined for each a in S. In the above context, for each x in X, we put

(1.1) M x = { a ∈ M | x • a is defined }.
Then Conditions (i), (ii), (iii) of Definition 1.7 imply

1 ∈ M x , ab ∈ M x ⇔ (a ∈ M x & b ∈ M x•a ), ∀ finite S ∃x(S ⊆ M x ).
A monoid action in the standard sense, i.e., an everywhere defined action, is a partial action. For a more typical case, consider the n-strand Artin braid monoid B + n . We recall that B + n is defined for n ∞ by the monoid presentation

(1.2) B + n = σ 1 , ... , σ n-1 σ i σ j = σ j σ i for |i -j| 2 σ i σ j σ i = σ j σ i σ j for |i -j| = 1 + .
Then we obtain a partial action of B + ∞ on N by putting

(1.3) n • a = n if a belongs to B + n , undefined otherwise.
A natural category can then be associated with every partial action of a monoid. Definition 1.8. For α a partial action of a monoid M on a set X, the category associated with α, denoted C(α), or C(M, X) if the action is clear, is defined by

Obj(C(M, X)) = X, Hom(C(M, X)) = {(x, a, x • a) | x ∈ X, a ∈ M }.
Example 1.9. We shall denote by B + the category associated with the action (1.3) of B + ∞ on N, i.e., we put

Obj(B + ) = N, Hom(B + ) = {(n, a, n) | n ∈ B + n }. Define ∆ : Obj(B + ) → Hom(B + ) by ∆(n) = (n, ∆ n , n). Then the well known fact that B +
n is a Garside monoid for each n [START_REF] Garside | The braid group and other groups[END_REF][START_REF] Kassel | Braid groups, Grad. Texts in Math[END_REF] easily implies that B + is a Garside category (as will be formally proved in Proposition 1.11 below).

The example of B + shows the benefit of going from a monoid to a category. The monoid B + ∞ is not a (left)-Garside monoid, because it is of infinite type and there cannot exist a global Garside element ∆. However, the partial action of (1.3) enable us to restrict to subsets B + n (submonoids in the current case) for which Garside elements exist: with the notation of (1.1), B + n is (B + ∞ ) n . Thus the categorical context allows to capture the fact that B + ∞ is, in some sense, locally Garside. It is easy to formalize these ideas in a general setting. Definition 1.10. Let M be a monoid with a partial action α on a set X. A sequence (∆ x ) x∈X of elements of M is called a left-Garside sequence for α if, for each x in X, the element x • ∆ x is defined and (LG ℓoc 3 ) Div(∆ x ) left-generates M x and a ∆ x implies ∆ x a ∆ x•a . The monoid M is said to be locally left-Garside with respect to α if it is left-preGarside and there is at least one left-Garside sequence for α.

A typical example of a locally left-Garside monoid is B + ∞ with its action (1.3) on N. Indeed, the sequence (∆ n ) n∈N is clearly a left-Garside sequence for (1.3).

The nexy result should appear quite natural.

Proposition 1.11. Assume that M is a locally left-Garside monoid with left-Garside sequence

(∆ x ) x∈X . Then C(M, X) is a left-Garside category with left-Garside map ∆ defined by ∆(x) = (x, ∆ x , x • ∆ x ).
Proof. By definition, (x, a, y) (x ′ , a ′ , y ′ ) in C(M, X) implies x ′ = x and a a ′ in M . So the hypothesis that M satisfies (LG 0 ) implies that C(M, X) does.

Next, (x, a, y)(y, b, z) = (x, a, y)(y, b ′ , z ′ ) implies ab = ab ′ in M , hence b = b ′ by (LG 1 ), and, therefore, C(M, X) satisfies (LG 1 ).

Assume (x, a, y)(y, b ′ , x ′ ) = (x, b, z)(z, a ′ , x ′ ) in Hom(C(M, X)). Then ab ′ = ba ′ holds in M . By (LG 2 ), a and b admit a right-lcm c, and we have a c, b c, and c ab ′ . By hypothesis, x • ab ′ is defined, hence so is x • c, and it is obvious to check that (x, c, x • c) is a right-lcm of (x, a, y) and (x, b, z) in Hom(C(M, X)). Hence C(M, X) satisfies (LG 2 ).

Assume that (x, a, y) is a nontrivial morphism of Hom(C(M, X)). This means that a is nontrivial, so, by (LG ℓoc 3 ), some left-divisor a ′ of ∆ x is a left-divisor of a. Then (x, a ′ , x • a ′ ) ∆(x) holds, and ∆(x) left-generates Hom(x, -).

Finally, assume (x, a, y) ∆(x) in Hom(C(M, X)). This implies a ∆ x in M . Then (LG ℓoc

3 ) in M implies ∆ x a∆ y . By hypothesis, y • ∆ y is defined, and we have (x, a, y)∆(y) = (x, a∆ y , x • a∆ y ), of which (x, ∆

x , x • ∆ x ) is a left-divisor in Hom(C(M, X)). So (LG 3 ) is satisfied in C(M, X).
It is not hard to see that, conversely, if M is a left-preGarside monoid, then C(M, X) being a left-Garside category implies that M is a locally left-Garside monoid. We shall not use the result here.

If M has a total action on X, i.e., if x • a is defined for all x and a, the sets M x coincide with M , and Condition (LG ℓoc 3 ) reduces to (LG 3 ). In this case, each element ∆ x is a left-Garside element in M , and M is a left-Garside monoid. A similar result holds for each set M x that turns out to be a submonoid (if any). Proposition 1.12. Assume that M is a locally left-Garside monoid with left-Garside sequence (∆ x ) x∈X , and x is such that M x is closed under product and ∆ y = ∆ x holds for each y in M x . Then M x is a left-Garside submonoid of M .

Proof. By definition of a partial action, x • 1 is defined, so M x contains 1, and it is a submonoid of M . We show that M x satisfies (LG 0 ), (LG 1 ), (LG 2 ), and (LG 3 ). First, a counter-example to (LG 0 ) in M x would be a counter-example to (LG 0 ) in M , hence M x satisfies (LG 0 ). Similarly, an equality ab = ab ′ with b = b ′ in M x would also contradict (LG 1 ) in M , so M x satisfies (LG 1 ). Now, assume that a and b admit in M x , hence in M , a common right-multiple c. Then a and b admit a right-lcm c ′ in M . By hypothesis, x • c is defined, and c ′ c holds. By definition of a partial action, x • c ′ is defined as well, i.e., c ′ lies in M x , and it is a right-lcm of a and b in M x . So M x satisfies (LG 2 ), and it is left-preGarside.

Next, ∆ x is a left-Garside element in M x . Indeed, let a be any nontrivial element of M x . By (LG ℓoc 3 ), there exists a nontrivial divisor a ′ of a satisfying a ′ ∆ x . By definition of a partial action, x • a ′ is defined, so it belongs to M x , and ∆ x leftgenerates M x . Finally, assume a ∆ x . As ∆ x belongs to M x , this implies a ∈ M x , hence ∆ x a∆ x•a by (LG ℓoc 3 ), i.e., ∆ x a∆ x since we assumed that the sequence ∆ is constant on M x . So ∆ x a left-Garside in M x .

Simple morphisms

We return to general left-Garside categories and establish a few basic results. As in the case of Garside monoids, an important role is played by the divisors of ∆, a local notion here.

2.1. Assume that C is a left-Garside category. A morphism f of C is called simple if it is a left-divisor of ∆(∂ 0 f ).
In this case, we denote by f * the unique simple morphism satisfying f f * = ∆(∂ 0 f ). The family of all simple morphisms in C is denoted by Hom sp (C).

By definition, every identity morphism 1 x is a left-divisor of every morphism with source x, hence in particular of ∆(x). Therefore 1 x is simple. Definition 2.2. Assume that C is a left-Garside category. We put φ(x) = ∂ 1 (∆(x)) for x in Obj(C), and φ(f ) = f * * for f in Hom sp (C).

Although straightforward, the following result is fundamental-and it is the main argument for stating (LG 3 ) in the way we did.

Lemma 2.3. Assume that C is a left-Garside category.

(i) If f is a simple morphism, so are f * and φ(f ).

(ii) Every right-divisor of a morphism ∆(x) is simple.

Proof. (i) By (LG 3 ), we have

f f * = ∆(∂ 0 f ) f ∆(∂ 1 f ), hence f * ∆(∂ 1 f
) by left-cancelling f . This shows that f * is simple. Applying the result to f * shows that φ(f )-as well as φ k (f ) for each positive k-is simple.

(ii) Assume that g is a right-divisor of ∆(x). This means that there exists f satisfying f g = ∆(x), hence g = f * by (LG 1 ). Then g is simple by (i) Lemma 2.4. Assume that C is a left-Garside category.

(i) The morphisms 1 x are the only left-or right-invertible morphisms in C.

(ii) Every morphism of C is a product of simple morphisms.

(iii) There is a unique way to extend φ into a functor of C into itself.

(iv) The map ∆ is a natural transformation of the identity functor into φ, i.e., for each morphism f , we have

(2.1) f ∆(∂ 1 f ) = ∆(∂ 0 f ) φ(f ).
Proof. (i) Assume f g = 1 x with f = 1 x and g = 1 ∂1f . Then we have

1 x ≺ f ≺ f g ≺ f ≺ f g ≺ ...,
an infinite ≺-increasing sequence in Div(1 x ) that contradicts (LG 0 ). (ii) Let f be a morphism of C, and let x = ∂ 0 f . If f is trivial, then it is simple, as observed above. We wish to prove that simple morphisms generate Hom(C). Owing to Lemma 1.2, it is enough to prove that simple morphisms left-generate Hom(C), i.e., that every nontrivial morphism with source x is left-divisible by a simple morphism with source x, in other words by a left-divisor of ∆(x). This is exactly what the first part of Condition (LG 3 ) claims.

(iii) Up to now, φ has been defined on objects, and on simple morphisms. Note that, by construction, (2.1) is satisfied for each simple morphism f . Indeed, applying Definition 2.1 for f and f * gives the relations

f f * = ∆(∂ 0 f ) and f * f * * = ∆(∂ 0 f * ) = ∆(∂ 1 f ), whence f ∆(∂ 1 f ) = f f * f * * = ∆(∂ 0 f )f * * = ∆(∂ 0 f )φ(f ).
Applying this to f = 1 x gives ∆(x) = ∆(x)φ(1 x ), hence φ(1 x ) = 1 φ(x) by (LG 1 ). Let f be an arbitrary morphism of C, and let f 1 ...f p and g 1 ...g q be two decompositions of f as a product of simple morphisms, which exist by (ii). Repeatedly applying (2.1) to f p , ..., f 1 and g q , ..., g 1 gives

f ∆(∂ 1 f ) = f 1 ...f p ∆(∂ 1 f ) = ∆(∂ 0 f )φ(f 1 )...φ(f p ) = g 1 ...g q ∆(∂ 1 f ) = ∆(∂ 0 f )φ(g 1 )...φ(g q ).
By (LG 1 ), we deduce φ(f 1 )...φ(f p ) = φ(g 1 )...φ(g q ), and therefore there is no ambiguity in defining φ(f ) to be the common value. In this way, φ is extended to all morphisms in such a way that φ is a functor and (2.1) always holds. Conversely, the above definition is clearly the only one that extends φ into a functor.

(iv) We have seen above that (2.1) holds for every morphism f , so nothing new is needed here. See Figure 1 for an illustration.

∆(x)

∆(y) Lemma 2.5. Assume that C is a left-Garside category. Then, for each object x and each simple morphism f , we have

φ(x) φ(y) φ(f ) f * y x f
(2.2) φ(∆(x)) = ∆(φ(x)) and φ(f * ) = φ(f ) * .
Proof. By definition, the source of ∆(x) is x and its target is φ(x), hence applying

(2.1) with f = ∆(x) yields ∆(x)∆(φ(x)) = ∆(x)φ(∆(x)), hence ∆(φ(x)) = φ(∆(x))
after left-cancelling ∆(x).

On the other hand, let x = ∂ 0 f . Then we have f f * = ∆(x), and ∂ 0 (φ(f )) = φ(x). Applying φ and the above relation, we find

φ(f )φ(f * ) = φ(∆(x)) = ∆(φ(x)) = ∆(∂ 0 (φ(f ))) = φ(f )φ(f ) * . Left-cancelling φ(f ) yields φ(f * ) = φ(f ) * .
Remark 2.6. We can now see that Definition 1.4 is equivalent to Definition 2.10 of [START_REF] Digne | Garside and locally Garside categories[END_REF]: the only difference is that, in the latter, the functor φ is part of the definition. Lemma 2.4(iv) shows that a left-Garside category in our sense is a left-Garside category in the sense of [START_REF] Digne | Garside and locally Garside categories[END_REF]. Conversely, the hypothesis that φ and ∆ satisfy (2.1) implies that, for f : x → y, we have ∆(x)φ(y) = f ∆(y), whence ∆(x) f ∆(y) and f * φ(y) = ∆(y), which, by (LG 1 ), implies φ(f ) = f * * . So every left-Garside category in the sense of [START_REF] Digne | Garside and locally Garside categories[END_REF] is a left-Garside category in the sense of Definition 1.4.

2.2.

The case of a locally left-Garside monoid. We now consider the particular case of a category associated with a partial action of a monoid M . Lemma 2.7. Assume that M is a locally left-Garside monoid with left-Garside sequence (∆ x ) x∈X . Then ∆ x a∆ x•a holds whenever x • a is defined, and, defining φ x (a) by ∆ x φ x (a) = a∆ x•a , we have

(2.3) φ(x) = x • ∆ x , φ((x, a, y)) = (φ(x), φ x (a), φ(y)).
Proof. Assume that x • a is defined. By Lemma 2.4(ii), the morphism (x, a, x • a) of C(M, X) can be decomposed into a finite product of simple morphisms (x 0 , a 1 , x 1 ), ..., , (x d-1 , a d , x d ). This implies a = a 1 ...a d in M . The hypothesis that each morphism

(x i-1 , a i , x i ) is simple implies ∆ xi-1 a i ∆ xi for each i, whence ∆ x a 1 ∆ x1 a 1 a 2 ∆ x2 ... a 1 ...a d ∆ x d = a∆ x•a .
Hence, for each element a in M x , there exists a unique element a ′ satisfying ∆ x a ′ = a∆ x•a , and this is the element we define to be φ x (a). Then,

x • a = y implies (x, a, y)(y, ∆ y , φ(y)) = (x, ∆ x , φ(x))(φ(x), φ x (a), φ(y)).
By uniqueness, we deduce φ((x, a, y)) = (φ(x), φ x (a), φ(y)).

Greatest common divisors.

We observe-or rather recall-that left-gcd's always exist in a left-preGarside category. We begin with a standard consequence of the noetherianity assumption (LG 0 ).

Lemma 2.8. Assume that C is a left-preGarside category and S is a subset of Hom(C) that contains the identity-morphisms and is closed under right-lcm. Then every morphism has a unique maximal left-divisor that lies in S.

Proof. Let f be an arbitrary morphism. Starting from f 0 = 1 ∂0f , which belongs to S by hypothesis, we construct a ≺-increasing sequence f 0 , f 1 , ... in S ∩ Div(f ).

As long as f i is not -maximal in S ∩ Div(f ), we can find f i+1 in S satisfying f i ≺ f i+1 f . Condition (LG 0 ) implies that the construction stops after a finite number d of steps. Then f d is a maximal left-divisor of f lying in S.

As for uniqueness, assume that g ′ and g ′′ are maximal left-divisors of f that lie in S. By construction, g ′ and g ′′ admit a common right-multiple, namely f , hence, by (LG 2 ), they admit a right-lcm g. By construction, g is a left-divisor of f , and it belongs to S since g ′ and g ′′ do. The maximality of g and g ′ implies g ′ = g = g ′′ . Proposition 2.9. Assume that C is a left-preGarside category. Then any two morphisms of C sharing a common source admit a unique left-gcd.

Proof. Let S be the family of all common left-divisors of f and g. It contains 1 ∂0f , and it is closed under lcm. A left-gcd of f and g is a maximal left-divisor of f lying in S. Lemma 2.8 gives the result.

Least common multiples.

As for right-lcm, the axioms of left-Garside categories only demand that a right-lcm exists when a common right-multiple does. A necessary condition for such a common right-multiple to exist is to share a common source. This condition is also sufficient. Again we begin with an auxiliary result.

Lemma 2.10. Assume that C is a left-Garside category. Then, for f = f 1 ...f d with f 1 , ..., f d simple and x = ∂ 0 f , we have (2.4) f ∆(x) ∆(φ(x)) ... ∆(φ d-1 (x)).
Proof. We use induction on d. For d = 1, this is the definition of simplicity. Assume

d 2. Put y = ∂ 1 f 1 . Applying the induction hypothesis to f 2 ...f d , we find f = f 1 (f 2 ...f d ) f 1 ∆(y) ∆(φ(y)) ... ∆(φ d-2 (y)) = ∆(x) ∆(φ(x)) ... ∆(φ d-2 (x)) φ d-1 (f 1 ) ∆(x) ∆(φ(x)) ... ∆(φ d-2 (x)) ∆(φ d-1 (x)).
The second equality comes from applying (2.1) d -1 times, and the last inequality comes from the fact that φ d-1 (f 1 ) is simple with source φ d-1 (x).

Proposition 2.11. Assume that C is a left-Garside category. Then any two morphisms of C sharing a common source admit a unique right-lcm.

Proof. Let f, g be any two morphisms with source x. By Lemma 2.4, there exists d such that f and g can be expressed as the product of at most d simple morphisms. Then, by Lemma 2.10, ∆(x) ∆(φ(x)) ... ∆(φ d-1 (x)) is a common right-multiple of f and g. Finally, (LG 2 ) implies that f and g admit a right-lcm. The uniqueness of the latter is guaranteed by Lemma 2.4(i).

In a general context of categories, right-lcm's are usually called push-outs (whereas left-lcm's are called pull-backs). So Proposition 2.11 states that every left-Garside category admits pushouts.

Applying the previous results to the special case of categories associated with a partial action gives analogous results for all locally left-Garside monoids.

Corollary 2.12. Assume that M is a locally left-Garside monoid with respect to some partial action of M on X.

(i) Any two elements of M admit a unique left-gcd and a unique right-lcm.

(ii) For each x in X, the subset M x of M is closed under right-lcm.

Proof. (i) As for left-gcd's, the result directly follows from Proposition 2.9 since, by definition, M is left-preGarside.

As for right-lcm's, assume that M is locally left-Garside with left-Garside sequence (∆ x ) x∈X . Let a, b be two elements of M . By definition of a partial action, there exists x in X such that both x

Regular left-Garside categories

The main interest of Garside structures is the existence of a canonical normal forms, the so-called greedy normal form [START_REF] Epstein | Word Processing in Groups[END_REF]. In this section, we adapt the construction of the normal form to the context of left-Garside categories-this was done in [START_REF] Digne | Garside and locally Garside categories[END_REF] already-and of locally left-Garside monoids. The point here is that studying the computation of the normal form naturally leads to introducing the notion of a regular left-Garside category, crucial in Section 6.

3.1. The head of a morphism. By Lemma 2.4(ii), every morphism in a left-Garside category is a product of simple morphisms. The decomposition need not to be unique in general, and the first step for constructing a normal form consists in isolating a particular simple morphism that left-divides the considered morphism. It will be useful to develop the construction in a general framework where the distinguished morphisms need not necessarily be the simple ones.

Notation. We recall that, for f, g in Hom(C), where C is a left-preGarside category, lcm(f, g) is the right-lcm of f and g, when it exists. In this case, we denote by f \g the unique morphism that satisfies

(3.1) f • f \g = lcm(f, g).
We use a similar notation in the case of a (locally) left-Garside monoid.

Definition 3.1. Assume that C is a left-preGarside category and S is included in Hom(C). We say that S is a seed for C if (i) S left-generates Hom(C), (ii) S is closed under the operations lcm and \, (iii) S is closed under left-divisor.

In other words, S is a seed for C if (i) every nontrivial morphism of C is leftdivisible by a nontrivial element of S, (ii) for all f, g in S, the morphisms lcm(f, g) and f \g belong to S whenever they exist, and (iii) for each f in S, the relation

h f implies h ∈ S. Lemma 3.2. If C is a left-Garside category, then Hom sp (C) is a seed for C.
Proof. First, Hom sp (C) left-generates Hom(C) by Condition (LG 3 ).

Next, assume that f, g are simple morphisms sharing the same source x. By Proposition 2.11, the morphisms lcm(f, g) and f \g exist. By definition, we have f ∆(x) and g ∆(x), hence lcm(f, g)

∆ x . Hence lcm(f, g) is simple. Let h satisfy lcm(f, g)h = ∆(x). This is also f (f \g) h = ∆(x). By Lemma 2.3(ii), (f \g) h, which is a right-divisor of ∆(x)
, is simple, and, therefore, f \g, which is a left-divisor of (f \g) h, is simple as well by transitivity of .

Finally, Hom sp (C) is closed under left-divisor by definition.

Lemma 2.8 guarantees that, if S is a seed for C, then every morphism f of C has a unique maximal left-divisor g lying in S, and Condition (i) of Definition 3.1 implies that g is nontrivial whenever f is. Definition 3.3. In the context above, the morphism g is called the S-head of f , denoted H S (f ).

In the case of Hom sp (C), it is easy to check, for each f in Hom(C), the equality

(3.2) H Hom sp (C) (f ) = gcd(f, ∆(∂ 0 f ));
in this case, we shall simply write H(f ) for H Hom sp (C) (f ).

Normal form.

The following result is an adaptation of a result that is classical in the framework of Garside monoids.

Proposition 3.4. Assume that C is a left-preGarside category and S is a seed for C. Then every nontrivial morphism f of C admits a unique decomposition

(3.3) f = f 1 ...f d ,
where f 1 , ..., f d lie in S, f d is nontrivial, and f i is the S-head of f i ...f d for each i.

Proof. Let f be a nontrivial morphism of C, and let f 1 be the S-head of f . Then f 1 belongs to S, it is nontrivial, and we have f = f 1 f ′ for some unique f ′ . If f ′ is trivial, we are done, otherwise we repeat the argument with f ′ . In this way we obtain a ≺-increasing sequence

1 ∂0f ≺ f 1 ≺ f 1 f 2 ≺ ... . Condition (LG 0 ) implies
that the construction stops after a finite number of steps, yielding a decomposition of the form (3.3). As for uniqueness, assume that (f 1 , ..., f d ) and (g 1 , ..., g e ) are decomposition of f that satisfy the conditions of the statement. We prove (f 1 , ..., f d ) = (g 1 , ..., g e ) using induction on min(d, e). First, d = 0 implies e = 0 by Lemma 2.4(i). Otherwise, the hypotheses imply f 1 = H S (f ) = g 1 . Left-cancelling f 1 gives two decompositions (f 2 , ..., f d ) and (g 2 , ..., g e ) of f 2 ...f d , and we apply the induction hypothesis. Definition 3.5. In the context above, the sequence (f 1 , ..., f d ) is called the Snormal form of f . When S turns out to be the family Hom sp (C), the S-normal form will be simply called the normal form. The interest of the S-normal form lies in that it is easily characterized and easily computed. First, one has the following local characterization of normal sequences. Proposition 3.6. Assume that C is a left-preGarside category and S is a seed for C. Then a sequence of morphisms (f 1 , ..., f d ) is S-normal if and only if each length two subsequence

(f i , f i+1 ) is S-normal.
This follows from an auxiliary lemma.

Lemma 3.7. Assume that (f 1 , f 2 ) is S-normal and g belongs to S. Then g f f 1 f 2 implies g f f 1 .
Proof. The hypothesis implies that f and g have the same source. Put g ′ = f \g. The hypothesis that S is closed under \ and an easy induction on the length of the S-normal form of f show that g ′ belongs to S. By hypothesis, we have both

g f f 1 f 2 and f f f 1 f 2 , hence lcm(f, g) = f g ′ f f 1 f 2 whence g ′ f 1 f 2 by left-cancelling f . As g belongs to S and (f 1 , f 2 ) is normal, this implies g ′ f 1 , and finally g f g ′ f f 1 .
Proof of Proposition 3.6. It is enough to consider the case d = 2, from which an easy induction on d gives the general case. So we assume that (f 1 , f 2 ) and (f 2 , f 3 ) are S-normal, and aim at proving that (f 1 , f 2 , f 3 ) is S-normal. The point is to prove that, if g belongs to S, then g f

1 f 2 f 3 implies g f 1 . So assume g f 1 f 2 f 3 . As (f 2 , f 3 ) is S-normal, Lemma 3.7 implies g f 1 f 2 . As (f 1 , f 2 ) is S-normal, this implies g f 1 .
3.3. A computation rule. We establish now a recipe for inductively computing the S-normal form, namely determining the S-normal form of gf when that of f is known and g belongs to S. Proposition 3.8. Assume that C is a left-preGarside category, S is a seed for C, and (f 1 , ..., f d ) is the S-normal form of f . Then, for each g in S, the S-normal form of gf is

(f ′ 1 , ..., f ′ d , g d ), where g 0 = g and (f ′ i , g i ) is the S-normal form of g i-1 f i for i increasing from 1 to d-see Figure 2.
Proof. For an induction, it is enough to consider the case d = 2, hence to prove Claim. Assume that the diagram

f 1 f 2 f ′ 1 f ′ 2 g 0 g 1 g 2 is commutative and (f 1 , f 2 ) and (f ′ 1 , g 1 ) are S-normal. Then (f ′ 1 , f ′ 2 ) is S-normal. f 1 f 2 f ′ 1 f ′ 2 g 0 g 1 g 2 ... ... g d-1 g d f d f ′ d Figure 2
. Adding one S-factor g0 on the left of an S-normal sequence (f1, ..., f d ): compute the S-normal form (f ′ 1 , g1) of g0f1, then the Snormal form (f ′ 2 , g2) of g1f2, and so on from left to right; the sequence

(f ′ 1 , ..., f ′ d , g d ) is S-normal.
So assume that h belongs to S and satisfies h f

′ 1 f ′ 2 . Then, a fortiori, we have h f ′ 1 f ′ 2 g 2 = g 0 f 1 f 2 , hence h g 0 f 1 by Lemma 3.7 since (f 1 , f 2 ) is S-normal. Therefore we have h f ′ 1 g 1 , hence h f ′ 1 since (f ′ 1 , g 1 ) is S-normal.
The results of Proposition 3.6 and 3.8 apply in particular when C is left-Garside and S is the family of all simple morphisms, in which case they involve the standard normal form.

In the case of lcm's, Corollary 2.12 shows how a result established for general left-Garside categories can induce a similar result for locally left-Garside monoids. The situation is similar with the normal form, provided some additional assumption is satisfied.

Definition 3.9. A left-Garside sequence (∆ x ) x∈X witnessing that a certain monoid is locally left-Garside is said to be coherent if, for all a, x, x ′ such that a•x is defined, a ∆ x ′ implies a ∆ x .
For instance, the family (∆ n ) n∈N witnessing for the locally left-Garside structure of the monoid B + ∞ is coherent. Indeed, a positive n-strand braid a is a left-divisor of ∆ n if and only if it is a left-divisor of ∆ n ′ for every n ′ n. The reason is that being simple is an intrinsic property of positive braids: a positive braid is simple if and only if it can be represented by a braid diagram in which any two strands cross at most once [START_REF] Elrifai | Algorithms for positive braids[END_REF]. Proposition 3.10. Assume that M is a locally left-Garside monoid associated with a coherent left-Garside sequence

(∆ x ) x∈X . Let Σ = {a ∈ M | ∃x ∈ X(a ∆ x )}.
Then Σ is a seed for M , every element of M admits a unique Σ-normal form, and the counterpart of Propositions 3.6 and 3.8 hold for the Σ-normal form in M .

Proof. Axiom (LG ℓoc

3 ) guarantees that every nontrivial element of M is left-divisible by some nontrivial element of Σ. Then, by hypothesis, 1 x ∆ x holds for each object x. Then, assume a, b ∈ Σ. There exists x such that x•a and x•b is defined. By definition of Σ, there exists x ′ satisfying a ∆ x ′ , hence, by definition of coherence, we have a ∆ x . A similar argument gives b ∆ x , whence lcm(a, b) ∆ x , and lcm(a, b) ∈ Σ. So there exists c satisfying a (a\b) c = ∆ x . By (LG ℓoc

3 ), we deduce (a\b) c ∆ x•a , whence a\b ∆ x•a , and we conclude that a\b belongs to Σ. Finally, it directly results from its definition that Σ is closed under left-divisor. Hence Σ is a seed for M in the sense of Definition 3.1.

As, by definition, M is a left-preGarside monoid, Proposition 3.4 applies, guaranteeing the existence and uniqueness of the Σ-normal form on M , and so do Propositions 3.6 and 3.8.

f ′ 1 f 1 g 0 g 1 ... ... g d-2 g d-1 g d f ′ d-1 f d-1 f ′ d f d
(g d-1 f ′ d ) of f d g d ), then the normal form (g d-2 f ′ d-2 ) of f d-1 g d-1
, and so on from right to left; the sequence

(g0, f ′ 1 , ..., f ′ d ) is normal.
Thus, the good properties of the greedy normal form are preserved when the assumption that a global Garside element ∆ exists is replaced by the weaker assumption that local Garside elements ∆ x exist, provided they satisfy some coherence.

3.4. Regular left-Garside categories. It is natural to look for a counterpart of the recipe of Proposition 3.8 involving right-multiplication by an element of the seed instead of left-multiplication. Such a counterpart exists but, interestingly, the situation is not symmetric, and we need a new argument. The latter demands that the considered category satisfies an additional condition, which is automatically satisfied in a two-sided Garside category, but not in a left-Garside category.

In this section, we only consider the case of a left-Garside category and its simple morphisms, and not the case of a general left-preGarside category with an arbitrary seed-see Remark 3.14. So, we only refer to the standard normal form. Definition 3.11. We say that a left-Garside category C is regular if the functor φ preserves normality of length 2 sequences: for

f 1 , f 2 simple with ∂ 1 f 1 = ∂ 0 f 2 , (3.4) (f 1 , f 2 ) normal implies (φ(f 1 ), φ(f 2 )) normal.
Proposition 3.12. Assume that C is a regular left-Garside category, that (f 1 , ..., f d ) is the normal form of a morphism f , and that g is simple. Then the normal form of f g is (g 0 , f ′ 1 , ..., f ′ d ), where g d = g and (g i-1 , f ′ i ) is the normal form of f i g i for i decreasing from d to 1-see Figure 3.

We begin with an auxiliary observation. Lemma 3.13. Assume that C is a left-Garside category and f 1 , f 2 are simple morphisms satisfying

∂ 1 f 1 = ∂ 0 f 2 . Then (f 1 , f 2 ) is normal if and only if f *
1 and f 2 are left-coprime, i.e., gcd(f * 1 , f 2 ) is trivial. Proof. The following equalities always hold:

H(f 1 f 2 ) = gcd(f 1 f 2 , ∆(∂ 0 f 1 )) = gcd(f 1 f 2 , f 1 f * 1 ) = f 1 gcd(f 2 , f * 1 ). Hence (f 1 , f 2 ) is normal, i.e., f 1 = H(f 1 f 2 ) holds, if and only if f 1 = f 1 gcd(f 2 , f * 1 ) does, which is gcd(f 2 , f * 1 ) = 1 ∂1f1
as left-cancelling f 1 is allowed. Proof of Proposition 3.12. As in the case of Proposition 3.8 it is enough to consider the case d = 2, and therefore it is enough to prove Claim. Assume that the diagram

f ′ 1 f ′ 2 f 1 f 2 g 0 g 1 g 2 is commutative and (f 1 , f 2 ) and (g 1 , f 2 ) are normal. Then (f ′ 1 , f ′ 2 ) is normal.
To prove the claim, we introduce the morphisms g * 0 , g * 1 , g * 2 defined by

g i g * i = ∆(∂ 0 g i ) (Definition 2.1). Then the diagram f ′ 1 f ′ 2 f 1 f 2 g 0 g 1 g 2 g * 0 g * 1 g * 2 φ(f 1 ) φ(f 2 )
is commutative. Indeed, appying (2.1), we find

g 0 f ′ 1 g * 1 = f 1 g 1 g * 1 = f 1 ∆(∂ 0 g 0 ) = f 1 ∆(∂ 1 f 1 ) = ∆(∂ 0 f 1 )φ(f 1 ) = ∆(∂ 0 g 0 )φ(f 1 ) = g 0 g * 0 φ(f 1 ), hence f ′ 1 g * 1 = g * 0 φ(f 1 ) by left-cancelling g 0 . A similar argument gives f ′ 2 g * 2 = g * 1 φ(f 2 ). Assume that h is simple and satisfies h f ′ 1 f ′ 2 . We deduce h f ′ 1 f ′ 2 g * 2 = g * 0 φ(f 1 )φ(f 2 ). By hypothesis, (f 1 , f 2 ) is normal. Hence the hypothesis that C is regular implies that (φ(f 1 ), φ(f 2 )) is normal as well. By Lemma 3.7, h g * 0 φ(f 1 )φ(f 2 ) implies h g * 0 φ(f 1 ) = f ′ 1 g * 1 . We deduce h gcd(f ′ 1 f ′ 2 , f ′ 1 g * 1 ) = f ′ 1 gcd(f ′ 2 , g * 1 )
. By Lemma 3.13, the hypothesis that (g 1 , f ′ 1 ) is normal implies gcd(g * 1 , f ′ 2 ) = 1, and, finally, we deduce h f ′ 1 , i.e., (f ′ 1 , f ′ 2 ) is normal. Remark 3.14. It might be tempting to mimick the arguments of this section in the general framework of a left-preGarside category C and a seed S, provided some additional conditions are satisfied. However, it is unclear that the extension can be a genuine one. For instance, if we require that, for each f in S, there exists f * in S such that f f * exists and depends on ∂ 0 f only, then the map ∂ 0 f → f f * is a left-Garside map and we are back to left-Garside categories.

3.5. Regularity criteria. We conclude with some sufficient conditions implying regularity. In particular, we observe that, in the two-sided case, regularity is automatically satisfied. Lemma 3.15. Assume that C is a left-Garside category C. Then a sufficient condition for C to be regular is that the functor φ is bijective on Hom(C).

Proof. Assume that C is a left-Garside category and φ is bijective on Hom(C). First we claim that φ(f ) φ(g) implies f g in C. Indeed, assume φ(g) = φ(f )h. As φ is surjective, we have φ(g) = φ(f )φ(h ′ ) for some h ′ , hence φ(g) = φ(f h ′ ) since φ is a functor, hence g = f h ′ since φ is injective.

Next, we claim that, if φ(f ) is simple if and only if f is simple. That the condition is sufficient directly follows from Definition 2.1. Conversely, assume that φ(f ) is simple. This means that there exists g satisfying φ(f )g = ∆(∂ 0 φ(f )). As φ is surjective, there exists g ′ satisfying g = φ(g ′ ). Applying (2.2), we obtain

φ(f g ′ ) = ∆(∂ 0 φ(f )) = φ(∆(∂ 0 f ), hence f g ′ = ∆(∂ 0 f ) by injectivity of φ.
Finally, assume that (f 1 , f 2 ) is normal, and g is a simple morphism left-dividing φ(f 1 )φ(f 2 ), hence satisfying gh = φ(f 1 )φ(f 2 ) for some h. As φ is surjective, we have g = φ(g ′ ) and h = φ(h ′ ) for some g ′ , h ′ . Moreover, by the claim above, the hypothesis that g is simple implies that g ′ is simple as well. Then we have φ

(g ′ )φ(h ′ ) = φ(f 1 )φ(f 2 ), hence g ′ h ′ = f 1 f 2 since φ is a functor and it is injective. The hypothesis that (f 1 , f 2 ) is normal implies g ′ f 1 , hence g = φ(g ′ ) φ(f 1 ). So (φ(f 1 ), φ(f 2 )
) is normal, and C is regular. Proposition 3.16. Every Garside category is regular.

Proof. Assume that C is a left-Garside with respect to ∆ and right-Garside with respect to ∇ satisfying ∆(x) = ∇(x ′ ) for x ′ = ∂ 1 ∆(x). Put ψ(x ′ ) = ∂ 0 ∇(x ′ ) for x ′ in Obj(C) and, for g simple in Hom(C), hence a right-divisor of ∇(∂ 1 g), denote by * g the unique simple morphism satisfying * gg = ∇(∂ 0 g), and put ψ(g) = * * g. Then arguments similar to those of Lemma 2.4 give the equality

(3.5) ∇(∂ 0 g) g = ψ(g) ∇(∂ 1 g)
which is an exact counterpart of (2.1). Let f : x → y be any morphism in C. Put x ′ = φ(x) and y ′ = φ(y). By construction, we also have x = ψ(x ′ ) and y = ψ(y ′ ). Applying (2.1) to f : x → y, we obtain ∆(x) φ(f ) = f ∆(y), which is also

(3.6) ∇(x ′ ) φ(f ) = f ∇(y ′ ).
On the other hand, applying (3.5) to φ(f ) :

x ′ → y ′ yields (3.7) ∇(x ′ ) φ(f ) = ψ(φ(f )) ∇(y ′ ).
Comparing (3.6) and (3.7) and right-cancelling ∇(y ′ ), we deduce ψ(φ(f )) = f . A symmetric argument gives φ(ψ(g)) = g for each g, and we conclude that ψ is the inverse of φ, which is therefore bijective. Then we apply Lemma 3.15.

Remark 3.17. The above proof shows that, if C is a left-Garside category that is Garside, then the associated functor φ is bijective both on Obj(C) and on Hom(C).

Let us mention without proof that this necessary condition is actually also sufficient.

Apart from the previous very special case, we can state several weaker regularity criteria that are close to the definition and will be useful in Section 6. We recall that H(f ) denoted the maximal simple morphism left-dividing f . Proposition 3.18. A left-Garside category C is regular if and only if φ preserves the head function on product of two simples: for f 1 , f 2 simple with

∂ 1 f 1 = ∂ 0 f 2 , (3.8) H(φ(f 1 f 2 )) = φ(H(f 1 f 2 ));
Proof. Assume that C is regular, and let

f 1 , f 2 satisfy ∂ 1 f 1 = ∂ 0 f 2 . Let (f ′ 1 , f ′ 2 ) be the formal form of f 1 f 2 -which has length 2 at most by Proposition 3.8. Then, (φ(f ′ 1 ), φ(f ′ 2 )) is normal and satisfies φ(f ′ 1 )φ(f ′ 2 ) = φ(f 1 f 2 ), so (φ(f ′ 1 ), φ(f ′ 2 )) is the normal form of f (f 1 f 2 ). Hence we have H(f 1 f 2 ) = f ′ 1 and H(φ(f 1 f 2 )) = φ(f ′ 1 ), which is (3.8).
Conversely, assume (3.8) and let (f 1 , f 2 ) be normal. By construction, we have

f 1 = H(f 1 f 2 ), hence φ(f 1 ) = H(φ(f 1 f 2 )) by hypothesis. This means that the normal form of φ(f 1 f 2 ) is (φ(f 1 ), g) for some g satisfying φ(f 1 f 2 ) = φ(f 1 )g. Now φ(f 2
) is such a morphism g, and, by (LG 1 ), it is the only one. So the normal form of φ(f 1 f 2 ) is (φ(f 1 ), φ(f 2 )), and C is regular. Proposition 3.19. Assume that C is a left-Garside category C. Then two sufficient conditions for C to be regular are (i) The functor φ preserves left-coprimeness of simple morphisms: for f, g simple with ∂ 0 f = ∂ 0 g, (3.9) gcd(f, g) = 1 implies gcd(φ(f ), φ(g)) = 1.

(ii) The functor φ preserves the gcd operation on simple morphisms: for f, g simple with ∂ 0 f = ∂ 0 g, (3.10) gcd(φ(f ), φ(g)) = φ(gcd(f, g)),

and, moreover, φ(f ) is nontrivial whenever f is nontrivial.

Proof. Assume (i). Let (f, g) be normal. By Lemma 3.13, we have gcd(f * , g) = 1. By (3.9), we deduce gcd(φ(f * ), φ(g)) = 1. By Lemma 2.5, this equality is also gcd(φ(f ) * , φ(g)) = 1, which, by Lemma 3.13 again, means that (φ(f ), φ(g)) is normal. Hence C is regular. On the other hand, it is clear that (ii) implies (i).

Self-distributivity

We quit general left-Garside categories, and turn to the description of one particular example, namely a certain category (two categories actually) associated with the left self-distributive law. The latter is the algebraic law (LD)

x(yz) = (xy)(xz) extensively investigated in [START_REF] Dehornoy | Braids and Self-Distributivity[END_REF]. We first review some basic results about this law and the associated free LDsystems, i.e., the binary systems that obey the LD-law. The key notion is the notion of an LD-expansion, with two derived categories LD + 0 and LD + that will be our main subject of investigation from now on.

4.1.

Free LD-systems. For each algebraic law (or family of algebraic laws), there exist universal objects in the category of structures that satisfy this law, namely the free systems. Such structures can be uniformly described as the quotient of some absolutely free structures under a convenient congruence. Definition 4.1. We let T n be the set of all bracketed expressions involving variables x 1 , ..., x n , i.e., the closure of {x 1 , ..., x n } under t 1 ⋆ t 2 = (t 1 )(t 2 ). We use T for the union of all sets T n . Elements of T are called terms.

Typical terms are x 1 , x 2 ⋆ x 1 , x 3 ⋆ (x 3 ⋆ x 1 ), etc. It is convenient to think of terms as rooted binary trees with leaves indexed by the variables: the trees associated with the previous terms are •

x 1 , x 2 x 1 , and

x 3 x 3 x 1
, respectively. The system (T n , ⋆)

is the absolutely free system (or algebra) generated by x 1 , ..., x n , and every binary system generated by n elements is a quotient of this system. So is in particular the free LD-system of rank n.

Definition 4.2. We denote by = LD the least congruence (i.e., equivalence relation compatible with the product) on (T n , ⋆) that contains all pairs of the form

(t 1 ⋆ (t 2 ⋆ t 3 ), (t 1 ⋆ t 2 ) ⋆ (t 1 ⋆ t 3 )).
Two terms t, t ′ satisfying t = LD t ′ are called LD-equivalent.

The following result is then standard.

Proposition 4.3. For each n ∞, the binary system (T n /= LD , ⋆) is a free LDsystem based on {x 1 , ..., x n }.

LD-expansions.

The relation = LD is a complicated object, about which many questions remain open. In order to investigate it, it proved useful to introduce the subrelation of = LD that corresponds to applying the LD-law in the expanding direction only.

Definition 4.4. Let t, t ′ be terms. We say that t ′ is an atomic LD-expansion of t, denoted t → 1 LD t ′ , if t ′ is obtained from t by replacing some subterm of the form t 1 ⋆ (t 2 ⋆ t 3 ) with the corresponding term (t 1 ⋆ t 2 ) ⋆ (t 1 ⋆ t 3 ). We say that t ′ is an LD-expansion of t, denoted t → LD t ′ , if there exists a finite sequence of terms t 0 , ..., t p satisfying t 0 = t, t p = t ′ , and t i-1 → 1 LD t i for 1 i p. By definition, being an LD-expansion implies being LD-equivalent, but the converse is not true. For instance, the term (x ⋆ x) ⋆ (x ⋆ x) is an (atomic) LD-expansion of x⋆(x⋆x), but the latter is not an LD-expansion of the former. However, it should be clear that = LD is generated by → LD , so that two terms t, t ′ are LD-equivalent if and only if there exists a finite zigzag t 0 , t 1 , ..., t 2p satisfying t 0 = t, t 2p = t ′ , and t i-1 → LD t i LD ← t i+1 for each odd i.

The first nontrivial result about LD-equivalence is that the previous zigzags may always be assumed to have length two. This result is similar to the property that, if a monoid M satisfies Ore's conditions-as the braid monoid B + n does for instance-then every element in the universal group of M can be expressed as a fraction of the form ab -1 with a, b in M . Proposition 4.5 plays a fundamental role in the sequel, and we need to recall some elements of its proof. Definition 4.6. [START_REF] Dehornoy | Free distributive groupoids[END_REF] First, a binary operation ⊙ ⋆ on terms is recursively defined by (4.1)

t ⊙ ⋆ x i = t ⋆ x i , t ⊙ ⋆ (t 1 ⋆ t 2 ) = (t ⊙ ⋆ t 1 ) ⋆ (t ⊙ ⋆ t 2 ).
Next, for each term t, the term φ(t) is recursively defined by2

(4.2) φ(x i ) = x i φ(t 1 ⋆ t 2 ) = φ(t 1 ) ⊙ ⋆ φ(t 2 ).
The idea is that t ⊙ ⋆ t ′ is obtained by distributing t everywhere in t ′ once. Then φ(t) is the image of t when ⋆ is replaced with ⊙ ⋆ everywhere in the unique expression of t in terms of variables. Examples are given in Figure 4. A straightforward induction shows that t ⊙ ⋆ t ′ is always an LD-expansion of t ⋆ t ′ and, therefore, that φ(t) is an LD-expansion of t. The main step for establishing Proposition 4.5 consists in proving that φ(t) plays with respect to atomic LD-expansions a role similar to Garside's fundamental 5. The fundamental LD-expansion φ(t) of a term t: an example;

φ = t 1 t 2 φ(t1) φ(t1) φ(t1) φ(t1) φ(t 2 )
φ = x 1 x 2 x 3 x 4 x 1 x 2 x 1 x 3 x 1 x 2 x 1 x 4 Figure
t = x1 ⋆ (x2 ⋆ (x3 ⋆ x4)) implies φ(t) = x1 ⊙ ⋆ (x2 ⊙ ⋆ (x3 ⊙ ⋆ x4)).
braid ∆ n with respect to Artin's generators σ i -which makes it natural to call φ(t) the fundamental LD-expansion of t. By construction, the category LD + 0 is left-and right-cancellative, and Proposiion 4.5 means that any two morphisms of LD + 0 with the same source admit a common right-multiple. Moreover, a natural candidate for being a left-Garside map is obtained by defining ∆(t) = (t, φ(t)) for each term t. Question 4.9. Is LD + 0 a left-Garside category? Question 4.9 is currently open. We shall see in Section 6.3 that it is one of the many forms of the so-called Embedding Conjecture. The missing part is that we do not know that least common multiples exist in LD + 0 , the problem being that we have no method for proving that a common LD-expansion of two terms is possibly a least common LD-expansion.

The monoid LD + and the category LD +

The solution for overcoming the above difficulty consists in developing a more precise study of LD-expansions that takes into account the position where the LDlaw is applied. This leads to introducing a certain monoid LD + whose elements provide natural labels for LD-expansions, and, from there, a new category LD + , of which LD + 0 is a projection. This category LD + is the one on which a left-Garside structure will be proved to exist.

5.1.

Labelling LD-expansions. By definition, applying the LD-law to a term t means selecting some subterm of t and replacing it with a new, LD-equivalent term. When terms are viewed as binary rooted trees, the position of a subterm can be specified by describing the path that connects the root of the tree to the root of

t D α t • D α α α t /α0 t /α10 t /α11
t /α0 t /α10 t /α0 t /α11 ↑ ↑ Figure 6. Action of Dα to a term t: the LD-law is applies to expand t at position α, i.e., to replace the subterm t /α , which is t /α0 ⋆ (t /α10 ⋆ t /α11 ), with (t /α0 ⋆ t /α10 ) ⋆ (t /α0 ⋆ t /α11 ); in other words, the light grey subtree is duplicated and distributed to the left of the dark grey and black subtrees.

the considered subtree, hence typically by a binary address, i.e., a finite sequence of 0's and 1's, according to the convention that 0 means "forking to the left" and 1 means "forking to the right". Hereafter, we use A for the set of all such addresses, and ǫ for the empty address, which corresponds to the position of the root in a tree.

Notation. For t a term and α an address, we denote by t /α the subtree of t whose root has address α, if it exists, i.e., if α is short enough. So, for instance, if t is the tree x 1 ⋆ (x 2 ⋆ x 3 ), we have t /0 = x 1 , t /10 = x 2 , whereas t /00 is not defined, and t /ǫ = t holds, as it holds for every term. Definition 5.1. (See Figure 6.) We say that t ′ is a D α -expansion of t, denoted t ′ = t • D α , if t ′ is the atomic LD-expansion of t obtained by applying LD at the position α, i.e., replacing the subterm t /α , which is t /0 ⋆ (t /10 ⋆ t /11 ), with the term (t /0 ⋆ t /10 ) ⋆ (t /0 ⋆ t /11 ).

By construction, every atomic LD-expansion is a D α -expansion for a unique α. The idea is to use the letters D α as labels for LD-expansions. As arbitrary LDexpansions are compositions of finitely many atomic LD-expansions, hence of D αexpansions, it is natural to use finite sequences of D α to label LD-expansions. In other words, we extend the (partial) action of D α on terms into a (partial) action of finite sequences of D α 's. Thus, for instance, we write

t ′ = t • D α D β D γ
to indicate that t ′ is the LD-expansion of t obtained by successively applying the LD-law (in the expanding direction) at the positions α, then β, then γ. Proof. Conditions (i) and (ii) of Definition 1.7 follow from the construction. The point is to prove (iii), i.e., to prove that, if w 1 , ..., w n are arbitrary finite sequences of letters D α , then there exists at least one term t such that t • w i is defined for each i. This is what [18, Proposition VII.1.21] states.

The monoid LD

+ . There exist clear connections between the action of various D α 's: different sequences may lead to the same transformations of trees. Our approach will consist in identifying a natural family of such relations and introducing the monoid presented by these relations. ("critical case") Sketch of proof. The commutation relation of the parallel case is clear, as the transformations involve disjoint subterms. The nested cases are commutation relations as well, but, because one of the involved subtree is nested in the other, it may be moved, and even possibly duplicated when the main expansion is performed, so that the nested expansion(s) correspond to different names before and after the main expansion. Finally, the critical case is specific to the LD-law, and there is no way to predict it except the verification, see Figure 7. For t a term and a in LD + , we shall naturally denote by t • a the common value of t • w for all sequences w of D α that represent a. Remark 5.6. In this way, each LD-expansion receives a label that is an element of LD + , thus becoming a labelled LD-expansion. However, we do not claim that a labelled LD-expansion are the same as an LD-expansion. Indeed, we do not claim that the relations of Lemma 5.3 exhaust all possible relations between the action of the D α 's on terms. A priori, it might be that different elements of LD + induce the same action on terms, so that one pair (t, t ′ ) might correspond to several labelled expansions with different labels. As we shall see below, the uniqueness of the labelling is another form of the above mentioned Embedding Conjecture.

D 1 D ǫ D 1 D 1 D ǫ D ǫ D 1 D 0

5.3.

The category LD + . We are now ready to introduce our main subject of interest, namely the category LD + of labelled LD-expansions. The starting point is the same as for LD + 0 , but the difference is that, now, we explicitly take into account the way of expanding the source is expanded into the target. Definition 5.7. We denote by LD + the category whose objects are terms, and whose morphisms are triples (t, a, t ′ ) with a in LD + and t • a = t ′ .

In other words, LD + is the category associated with the partial action of LD + on terms, in the sense of Section 1.8. We recall our convention that, when the morphisms of a category are triples, the source is the first entry, and the target is the the last entry. So, for instance, a typical morphism in LD + is the triple

, D ǫ D 1 , , ,
whose source is the term x ⋆ (x ⋆ (x ⋆ x)) (we the default convention that unspecified variables means some fixed variable x), and whose target is the term (x ⋆ x) ⋆ ((x ⋆ x) ⋆ (x ⋆ x)).

5.4.

The element ∆ t . We aim at proving that the category LD + is a left-Garside category. To this end, we need to define the ∆-morphisms. As planned in Section 4.3, the latter will be constructed using the LD-expansions (t, φ(t)). Defining a labelled version of this expansion means fixing some canonical way of expanding a term t into the corresponding term φ(t). A natural solution then exists, namely following the recursive definition of the operations ⊙ ⋆ and φ.

For w a word in the letters D α , we denote by sh 0 (w) the word obtained by replacing each letter D α of w with the corresponding letter D 0α , i.e., by shifting all indices by 0. Similarly, we denote by sh γ (w) the word obtained by appending γ on the left of each address in w. The LD-relations of Lemma 5.3 are invariant under shifting: if w and w ′ represent the same element a of LD + , then, for each γ, the words sh γ (w) and sh γ (w ′ ) represent the same element, naturally denoted sh γ (a), of LD + . By construction, sh γ is an endomorphism of the monoid LD + . For each a in LD + , the action of sh γ (a) on a term t corresponds to the action of a to the γ-subterm of t: so, for instance, if t ′ = t • a holds, then t ′ ⋆ t 1 = (t ⋆ t 1 ) • sh 0 (a) holds as well, since the 0-subterm of t ⋆ t 1 is t, whereas that of t ′ ⋆ t 1 is t ′ . Definition 5.8. For each term t, the elements δ t and ∆ t of LD + are defined by the recursive rules δ t = 1 for t of size 1, i.e., when t is a variable

x i , D ǫ • sh 0 (δ t0 ) • sh 1 (δ t1 ) for t = t 0 ⋆ t 1 . (5.1) ∆ t = 1 for t of size 1, sh 0 (∆ t0 ) • sh 1 (∆ t1 ) • δ φ(t1) for t = t 0 ⋆ t 1 . (5.2)
Example 5.9. Let t be x⋆(x⋆(x⋆x)). Then t /0 is x, and, therefore, ∆

t /0 is 1. Next, t /1 is x ⋆ (x ⋆ x), so (5.2) reads ∆ t = sh 1 (∆ t /1 ) • δ φ(t /1 ) . Then φ(t /1 ) is (x ⋆ x) ⋆ (x ⋆ x).
Applying (5.1), we obtain

δ φ(t /1 ) = D ǫ • sh 0 (δ x⋆x ) • sh 1 (δ x⋆x ) = D ǫ D 0 D 1 .
On the other hand, using (5.2) again, we fing

∆ t /1 = sh 0 (∆ x ) • sh 1 (∆ x⋆x ) • δ x⋆x = 1 • 1 • D ǫ = D ǫ ,
and, finally, we obtain ∆ t = D 1 D ǫ D 0 D 1 . According to the defining relations of the monoid LD + , this element is also D ǫ D 1 D ǫ . Note the compatibility of the result with the examples of Figures 5 and7.

Lemma 5.10. For all terms t 0 , t, we have

(t 0 ⋆ t) • δ t = t 0 ⊙ ⋆ t, (5.3) t • ∆ t = φ(t). (5.4)
The proof is an easy inductive verification. 5.5. Connection with braids. Before investigating the category LD + more precisely, we describe the simple connection existing between the category LD + and the positive braid category B + of Example 1.9.

Lemma 5.11. Define π :

{D α | α ∈ A} → {σ i | i 1} ∪ {1} by (5.5) π(D α ) = σ i+1 if α is the address 1 i , i.e., 11...1, i times 1, 1 otherwise.
Then π induces a surjective monoid homomorphism of LD + onto B + ∞ . Proof. The point is that each LD-relation of Lemma 5.3 projects under π onto a braid equivalence. All relations involving addresses that contain at least one 0 collapse to mere equalities. The remaining relations are

D 1 i D 1 j = D 1 j D 1 i with j i + 2,
which projects to the valid braid relation σ i-1 σ j-1 = σ j-1 σ i-1 , and

D 1 i D 1 j D 1 i = D 1 j D 1 i D 1 j D 1 i 0 , with j = i + 1, which projects to the not less valid braid relation σ i-1 σ j-1 σ i-1 = σ j-1 σ i-1 σ j-1 .
We introduced a category C(M, X) for each monoid M partially acting on X in Definition 1.8. The braid category B + and our current category LD + are of these type. For such categories, natural functors arise from morphisms between the involved monoids, and we fix the following notation. Definition 5.12. Assume that M, M ′ are monoids acting on sets X and X ′ , respectively. A morphism ϕ: M →M ′ and a map ψ: X→X ′ are called compatible if

(5.6) ψ(x • a) = ψ(x) • ϕ(a)
holds whenever x • a is defined. Then, we denote by [ϕ, ψ] the functor of C(M, X) to C(M ′ , X ′ ) that coincides with ψ on objects and maps (x, a, y) to (ψ(x), ϕ(a), ψ(y)).

Proposition 5.13. Define the right-height ht(t) of a term t by ht(x i ) = 0 and ht(t 0 ⋆ t 1 ) = ht(t 1 ) + 1. Then the morphism π of (5.5) is compatible with ht, and [π, ht] is a surjective functor of LD + onto B + .

The parameter ht(t) is the length of the rightmost branch in t viewed as a tree or, equivalently, the number of final )'s in t viewed as a bracketed expression.

Proof. Assume that (t, a, t ′ ) belongs to Hom(LD + ). Put n = ht(t). The LD-law preserves the right-height of terms, so we have ht(t ′ ) = n as well. The hypothesis that t • a exists implies that the factors D 1 i that occur in some (hence in every) expression of a satisfy i < n-1. Hence π(a) is a braid of B + n , and n•π(a) is defined. Then the compatibility condition (5.6) is clear, [π, ht] is a functor of LD + to B + . Surjectivity is clear, as each braid σ i belongs to the image of π.

Moreover, a simple relation connects the elements ∆ t of LD + and the braids ∆ n .

Proposition 5.14. We have π(∆ t ) = ∆ n whenever t has right-height n 1.

Proof. We first prove that ht(t) = n implies (5.7) π(δ t ) = σ 1 σ 2 ...σ n using induction on the size of t. If t is a variable, we have ht(t) = 0 and δ t = 1, so the equality is clear. Otherwise, write t = t 0 ⋆ t 1 . By definition, we have

δ t = D ǫ • sh 0 (δ t0 ) • sh 1 (δ t1 ).
Let sh denote the endomorphism of B + ∞ that maps σ i to σ i+1 for each i. Then π collapses every term in the image of sh 0 , and π(sh 1 (a)) = sh(π(a)) holds for each a in LD + . Hence, using the induction hypothesis π(δ t1 ) = σ 1 ...σ n-1 , we deduce

π(δ t ) = σ 1 • 1 • sh(σ 1 ...σ n-1 ) = σ 1 ...σ n ,
which is (5.7). Put ∆ 0 = 1 (= ∆ 1 ). We prove that ht(t) = n implies π(∆ t ) = ∆ n for n 0, using induction on the size of t again. If t is a variable, we have n = 0 and ∆ t = 1, as expected. Otherwise, write t = t 0 ⋆ t 1 . The definition gives

∆ t = sh 0 (∆ t0 ) • sh 1 (∆ t1 ) • δ φ(t1) .
As above, π collapses the term in the image of sh 0 , and it transforms sh 1 into sh. Hence, using the induction hypothesis π(∆ t1 ) = ∆ n-1 and (5.7) for φ(t 1 ), whose right-height is that of t 1 , we obtain

π(∆ t ) = 1 • sh(∆ n-1 ) • σ 1 σ 2 ... σ n-1 = ∆ n .

The main results

We can now state the two main results of this paper. Theorem 6.1. For each term t, put ∆(t) = (t, ∆ t , φ(t)). Then LD + is a left-Garside category with left-Garside map ∆, and [π, ht] is a surjective right-lcm preserving functor of LD + onto the positive braid category B + . Theorem 6.2. Unless the category LD + is not regular, the Embedding Conjecture of [18, Chapter IX] is true.

6.1.

Recognizing left-preGarside monoids. Owing to Proposition 1.11 and to the construction of LD + from the partial action of the monoid LD + on terms, the first part of Theorem 6.1 is a direct consequence of Proposition 6.3. The monoid LD + equipped with its partial action on terms via self-distributivity is a locally left-Garside monoid with associated left-Garside sequence (∆ t ) t∈T . This is the result we shall prove now. The first step is to prove that LD + is left-preGarside. To do it, we appeal to general tools that we now describe. As for (LG 0 ), we have an easy sufficient condition when the action turns out to be monotonous in the following sense. Proposition 6.4. Assume that M has a partial action on X and there exists a map µ : X → N such that a = 1 implies µ(x • a) > µ(x). Then M satisfies (LG 0 ).

Proof. Assume that (a 1 , ..., a ℓ ) is a ≺-increasing sequence in Div(a). By definition of a partial action, there exists x in X such that x • a is defined, and this implies that x • a i is defined for each i. Next, the hypothesis that (a 1 , ..., a ℓ ) is ≺-increasing implies that there exist b 2 , ..., b ℓ = 1 satisfying a i = a i-1 b i for each i. We find

µ(x • a i ) = µ((x • a i-1 ) • b i ) > µ(x • a i-1 ),
and the sequence (µ(x • a 1 ), ..., µ(x • a ℓ )) is increasing. As µ(x • a 1 ) µ(x) holds, we deduce ℓ µ(x • a) -µ(x) + 1 and, therefore, M satisfies (LG 0 ).

As for conditions (LG 1 ) and (LG 2 ), we appeal to the subword reversing method of [START_REF] Dehornoy | Complete positive group presentations[END_REF]. If S is any set, we denote by S * the set of all finite sequences of elements of S, i.e., of words on the alphabet S. Then S * equipped with concatenation is a free monoid. We use ǫ for the empty word. 6.2. Proof of Therem 6.1. We shall now prove that the monoid LD + equipped its partial action on terms via left self-distributivity satisfies the criteria of Section 6.1.

C(u, v 1 v 2 ) = C(u, v 1 ) C( C(v 1 , u), v 2 ), C(v 1 v 2 , u) = C(v 2 , C(u, v 1 )).
Here, and in most subsequent developments, we heavily appeal to the results of [START_REF] Dehornoy | Braids and Self-Distributivity[END_REF], some of which have quite intricate proofs.

Proof of Theorem 6.1. First, each term t has a size µ(t), which is the number of inner nodes in the associated binary tree. Then the hypothesis of Proposition 6.4 clearly holds: if t ′ is a nontrivial LD-expansion of t, then the size of t ′ is larger than that of t. Then, by Proposition 6.4, LD + satisfies (LG 0 ). Next, we observe that the presentation of LD + in Definition 5.4 is associated with a complement on the set {D α | α ∈ A}. Indeed, for each pair of addresses α, β, there exists in the list R LD exactly one relation of the type D α ... = D β .... Hence, in view of Proposition 6.6, and because we know that LD + satisfies (LG 0 ), it suffices to check that (6.2) holds in LD + for each triple D α , D β , D γ . This is Proposition VIII.1.9 of [START_REF] Dehornoy | Braids and Self-Distributivity[END_REF]. Hence LD + satisfies (LG 1 ) and (LG 2 ), and it is a left-preGarside monoid.

Let us now consider the elements ∆ t of Definition 5.8. First, by Lemma 5.10, t • ∆ t is defined for each term t, and it is equal to φ(t). Next, assume that t • D α is defined. Then Lemma VII.3.16 of [START_REF] Dehornoy | Braids and Self-Distributivity[END_REF] states that D α is a left-divisor of ∆ t in LD + , whereas Lemma VII.3.17 of [START_REF] Dehornoy | Braids and Self-Distributivity[END_REF] states that ∆ t is a left-divisor of D α ∆ t•Dα . Hence Condition (LG ℓoc 3 ) of Definition 1.10 is satisfied, and the sequence (∆ t ) t∈T is a left-Garside sequence in LD + . Hence LD + is a locally left-Garside monoid, which completes the proof of Proposition 6.3. By Proposition 1.11, we deduce that LD + , which is C(LD + , T ) by definition, is a left-Garside category with left-Garside map ∆ as defined in Theorem 6.1.

As for the connection with the braid category B + , we saw in Proposition 5.13 that [π, ht] is a surjective functor of LD + onto B + , and it just remains to prove that it preserves right-lcm's. This follows from the fact that the homomorphism π of LD + to B + ∞ preserves right-lcm's, which in turn follows from the fact that LD + and B + ∞ are associated with complements C and C satisfying, for each pair of addresses α, β,

(6.3) π(C(D α , D β )) = C(π(D α ), π(D β )).
Indeed, let a, b be any two elements of LD + . Let u, v be words on the alphabet {D α | α ∈ A} that represent a and b, respectively. By Proposition II.2.16 of [START_REF] Dehornoy | Braids and Self-Distributivity[END_REF], the word C(u, v) exists, and u C(u, v) represents lcm(a, b). Then π(u C(u, v)) represents a common right-multiple of the braids π(a) and π(b), and, by (6.3), we have

π(u C(u, v)) = π(u) C(π(u), π(v)).
This shows that the braid represented by π(u C(u, v)), which is π(lcm(a, b)) by definition, is the right-lcm of the braids π(a) and π(b). So the morphism π preserves right-lcm's, and the proof of Theorem 6.1 is complete.

6.3. The Embedding Conjecture. From the viewpoint of self-distributive algebra, the main benefit of the current approach might be that its leads to a natural program for possibly establishing the so-called Embedding Conjecture. This conjecture, at the moment the most puzzling open question involving free LD-systems, can be stated in several equivalent forms. Proposition 6.7. [18, Section IX.6] The following are equivalent: (i) The monoid LD + embeds in a group;

(ii) The monoid LD + admits right-cancellation;

(iii) The categories LD + 0 and LD + are isomorphic; (iv) The functor φ associated with the category LD + is injective; (v) For each term t, the LD-expansions of t make an upper-semilattice; (vi) The relations of Lemma 5.3 generate all relations that connect the action of D α 's by self-distributivity. Each of the above properties is conjectured to be true: this is the Embedding Conjecture.

We turn to the proof of Theorem 6.2. So our aim is to show that the Embedding Conjecture is true whenever the category LD + is regular. To this end, we shall use some technical results from [START_REF] Dehornoy | Braids and Self-Distributivity[END_REF], plus the following criterion, which enables one to prove right-cancellability by only using simple morphisms. Proposition 6.8. Assume that C is a left-Garside category and the associated functor φ is injective on Obj(C). Then the following are equivalent:

(i) Hom(C) admits right-cancellation;

(ii) The functor φ is injective on Hom(C). Moreover, if C is regular, (i) and (ii) are equivalent to (iii) The functor φ is injective on simple morphisms of C.

if LD + is regular, then Hom(LD + ) admits right-cancellation, which is one of the forms of the Embedding Conjecture, namely (ii) in Proposition 6.7.

6.4.

A program for proving the regularity of LD + . At this point, we are left with the question of proving (or disproving) Conjecture 6.11. The left-Garside category LD + is regular.

The regularity criteria of Section 3.5 lead to a natural program for possibly proving Conjecture 6.11 and, therefore, the Embedding Conjecture.

We begin with a preliminary observation.

Lemma 6.12. The left-Garside sequence (∆ t ) t∈T on LD + is coherent (in the sense of Definition 3.9).

Proof. The question is to prove that, if t is a term and t • a is defined and a ∆ t ′ holds for some t ′ , then we necessarily have a ∆ t . This is a direct consequence of Proposition VIII.5.1 of [START_REF] Dehornoy | Braids and Self-Distributivity[END_REF]. Indeed, the latter states that an element a is a left-divisor of some element ∆ t if and only if a can be represented by a word in the letters D α that has a certain special form. This property does not involve the term t, and it implies that, if a left-divides ∆ t , then it automatically left-divides every element ∆ t ′ such that t ′

• a is defined. So, according to Proposition 3.10, we obtain a well defined notion of a simple element in LD + : an element a of LD + is called simple if it left-divides at least one element of the form ∆ t . Then simple elements form a seed in LD + , and are eligible for a normal form satisfying the general properties described in Section 3. In this context, applying Proposition 3.19(ii) leads to the following criterion. Proposition 6.13. Assume that, for each term t and all simple elements a, b of LD + such that t • a and t • b are defined, we have (6.4) gcd(φ t (a), φ t (b)) = φ t (gcd(a, b)).

Then Conjecture 6.11 is true.

Proof. Let f, g be two simple morphisms in LD + that satisfy ∂ 0 f = ∂ 0 g = t. By definition, f has the form (t, a, t • a) for some a satisfying a ∆ t , hence simple in LD + . Similarly, f has the form (t, b, t • b) for some simple element b, and we have gcd(f, g) = (t, gcd(a, b), t • gcd(a, b)). On the other hand, Lemma 2.7 gives φ(f ) = (φ(t), φ t (a), φ(t • a)) and. φ(g) = (φ(t), φ t (b), φ(t • b)), whence gcd(φ(f ), φ(g)) = (φ(t), gcd(φ t (a), φ t (b)), φ(t) • gcd(φ t (a), φ t (b))).

If (6.4) holds, we deduce gcd(φ(f ), φ(g)) = φ(gcd(f, g)).

By Proposition 3.19(ii), this implies that LD + is regular.

(t) = ((x ⋆ x) ⋆ (x ⋆ x)) ⋆ ((x ⋆ x) ⋆ (x ⋆ x)).
An easy computation gives φ t (D ǫ ) = D 0 D 1 and φ t (D 1 ) = D ǫ , see Figure 8. We find gcd(φ t (a), φ t (b)) = 1 = gcd(a, b), and (6.4) is true in this case.

Note that the couterpart of (6.4) involving right-lcm's fails. In the current case, we have lcm(φ t (a), the terms φ(t • D ǫ ) and φ(t • D 1 ) admit a common LD-expansion that is smaller than φ t (t • lcm(D ǫ , D 1 )), which turns out to be φ 2 (t), see Figure 8 again. The reader may similarly check that (6. α denotes D α D α1 ...D α1 e-1 and > refers to the unique linear ordering of A satisfying α > α0β > α1γ for all α, β, γ. In this way, we associate with every simple element a of LD + a sequence of nonnegative integers (e α ) α∈A that plays the role of a sequence of coordinates for a. Then it should be possible to -express the coordinates of φ t (a) in terms of those of a, -express the coordinates of gcd(a, b) in terms of those of a and b. If this were done, proving (or disproving) the equalities (6.4) should be easy.

φ t (b)) = φ t (lcm(a, b)) • D 0 D 1 : t t • D 1 φ(t) φ(t • D 1 ) D 1 D ǫ D ǫ D ǫ D 0 D 1 D 0 D 1 D 00 D 10 D 1 D ǫ t • D ǫ t•lcm(D ǫ ,D 1 ) φ(t•D ǫ ) φ(t)•lcm(D 0 D 1 ,D ǫ ) φ(t•lcm(D ǫ ,D 1 )) D 1 D ǫ D ǫ D 1 D ǫ =
Remark. Contrary to the braid relations, the LD-relations of Lemma 5.3 are not symmetric. However, it turns out that the presentation of LD + is also associated with what can naturally be called a left-complement, namely a counterpart of a (right)-complement involving left-multiples. But the latter fails to satisfy the counterpart of (6.2), and it is extremely unlikely that one can prove that the monoid LD + is possibly right-cancellative (which would imply the Embedding Conjecture) using some version of Proposition 6.6.

Reproving braids properties

Proposition 5.13 and Theorem 6.1 connect the Garside structures associated with self-distributivity and with braids, both being previously known to exist. In this section, we show how the existence of the Garside structure of braids can be (re)-proved to exist assuming the existencce of the Garside structure of LD + only. So, for a while, we pretend that we do not know that the braid monoid B + n has a Garside structure, and we only know about the Garside structure of LD + . 7.1. Projections. We begin with a general criterion guaranteeing that the projection of a locally left-Garside monoid is again a locally left-Garside monoid.

If π is a map of S to S * , we still denote by π the alphabetical homomorphism of S * to S * that extends π, defined by π(s 1 ...s ℓ ) = π(s 1 )...π(s ℓ ).

Lemma 7.1. Assume that • M is a locally left-Garside monoid associated with a complement C on S;

• M is a monoid associated with a complement C on S and satisfying (LG 0 );

• π : S → S ∪ {ǫ} satisfies π(S) ⊇ S and Then M is left-preGarside, and π induces a surjective right-lcm preserving homomorphism of M onto M .

Proof. An easy induction shows that, if u, v are words on S and C(u, v) exists, then C(π(u), π(v)) exists as well and we have in M , which shows that the homomorphism of S * to M that extends π induces a well defined homomorphism of M to M . This homomorphism, still denoted π, is surjective since, by hypothesis, its image includes S.

(7.2) C(π(u), π(v)) = π( C(u, v)).
Finally, we claim that π preserves right-lcm's. The argument is almost the same as in the proof of Theorem 6.1, with the difference that, here, we do not assume that common multiples necessarily exist. Let a, b be two elements of M that admit a common right-multiple. Let u, v be words on S * that represent a and b, respectively. By Proposition II.2.16 of [START_REF] Dehornoy | Braids and Self-Distributivity[END_REF], the word C(u, v) exists, and u C(u, v) represents lcm(a, b). Then the word π(u C(u, v)) represents a common right-multiple of π(a) and π(b) in M , and, by (7.2), we have

π(u C(u, v)) = π(u) C(π(u), π(v)),
which shows that the element represented by π(u C(u, v)), which is π(lcm(a, b)) by definition, is the right-lcm of π(a) and π(b) in M .

We turn to locally left-Garside monoids, i.e., we add partial actions in the picture. Although lengthy, the following result is easy. It just says that, if M is a locally left-Garside monoid, then its image under a projection that is compatible with the various ingredients of the Garside structure is again locally left-Garside.

Proposition 7.2. Assume that

• M is a locally left-Garside monoid associated with a complement C on S and (∆ x ) x∈X is a left-Garside sequence for the involved action of M on X;

• M is a monoid associated with a complement C on S that has a partial action on X and satisfies (LG 0 );

• π : S → S ∪ {ǫ} satisfies (7.2), θ : S → S is a section for π, ̟ : X → X is a surjection, and

(7.3) For x in X and a in M , if x • a is defined, then so is ̟(x) • π(a)
and we have ̟(x) • π(a) = ̟(x • a);

(7.4) For x in X and a in S, if x • a is defined, then so is x • θ(a) for each x satisfying ̟(x) = x; (7.5) For x in X, the value of π(∆ x ) depends on ̟(x) only.

For x in X, let ∆ x be the common value of π(∆ x ) for ̟(x) = x. Then M is locally left-Garside, with associated left-Garside sequence (∆ x ) x∈X , and π induces a surjective right-lcm preserving homomorphism of M onto M .

Proof. First, the hypotheses of Lemma 7.1 are satisfied, hence M is left-preGarside and π induces a surjective lcm-preserving homomorphism of M onto M .

Next, by (7.5), the definition of the elements ∆ x for x in X is unambiguous. It remains to check that the sequence (∆ x ) x∈X is a left-Garside sequence with respect to the action of M on X. So, assume x ∈ M , and let x be any element of M satisfying ̟(x) = x.

First, x • ∆ x is defined, hence, by (7.3), so is ̟(x) • π(∆ x ), which is x • ∆ x . Assume now a = 1 and x • a is defined. As S generates M , we can assume a ∈ S without loss of generality. By (7.4), the existence of x • a implies that of x • θ(a). As (∆ x ) x∈X is a left-Garside sequence for the action of M on X, we have a ′ ∆ x for some a ′ = 1 left-dividing θ(a). By construction, θ(a) lies in S, and it is an atom in M . So the only possiblity is a ′ = θ(a), i.e., we have θ(a) ∆ x . Applying π, we deduce a ∆ x in M .

Finally, under the same hypotheses, we have ∆

x θ(a) ∆ x•θ(a) in M . Using π(∆ x•θ(a) ) = ∆ ̟(x•θ(a)) = ∆ ̟(x)•a = ∆ x•a ,
we deduce ∆ x a ∆ x•a in M , always under the hypothesis a ∈ S. The case of an arbitrary element a for which x • a exists then follows from an easy induction on the length of an expression of a as a product of elements of S.

It should then be clear that, under the hypotheses of Proposition 7.2, [π, ̟] is a surjective, right-lcm preserving functor of C(M, X) to C(M , X). ∞ is a locally left-Garside monoid with respect to its action on N, and (∆ n ) n∈N is a left-Garside sequence in B + ∞ . Proof. Hereafter, we denote by C the complement on {D α | α ∈ A} associated with the LD-relations of Lemma 5.3, and by C the complement on {σ i | i 1} associated with the braid relations of (1.2). We consider the maps π of Lemma 5.11, and ht from terms to nonnegative integers. Finally, we define θ by θ(σ i ) = D 1 i-i . We claim that these data satisfy all hypotheses of Proposition 7.2. The verifications are easy. That the complements C and C satisfy (7.1) follows from a direct inspection. For instance, we find

π(C(D 1 , D ǫ )) = π(D ǫ D 1 D 0 ) = σ 1 σ 2 = C(σ 2 , σ 1 ) = C(π(D 1 ), π(D ǫ )),
and similar relations hold for all pairs of generators D α , D β .

Then, the action of LD + on terms preserve the right-height, whereas the action of braids on N is trivial, so (7.3) is clear. Next, define θ by θ(σ i ) = D 1 i-1 . Then θ is a section for π, and we observe that t • θ(σ i ) is defined if and only if the right-height of t is at least i + 1, hence if and only if ht(t) • σ i is defined, so (7.4) is satisfied. Finally, we observed in Proposition 5.13 that π(∆ t ) is equal to ∆ ht(t) , hence it depends on ht(t) only. So (7.5) is satisfied.

Therefore, Proposition 7.2 applies, and it gives the expected result.

Corollary 7.4. (i) The braid category B + is a left-Garside category.

(ii) For each n, the braid monoid B + n is a Garside monoid. Proof. Point (i) follows from Proposition 1.11 once we know that B + ∞ is locally left-Garside. Point (ii) follows from Proposition 1.12 since, for each n, the submonoid B + n is the monoid (B + ∞ ) n in the sense of Definition 1.7. Thus, as announced, the Garside structure of braids can be recovered from the left-Garside structure associated with the LD-law.

Intermediate categories

We conclude with a different topic. The projection of the self-distributivity category LD + to the braid category B + described above is partly trivial in that terms are involved through their right-height only and the corresponding action of braids on integers is just constant. Actually, one can consider alternative projections corresponding to less trivial braid actions and leading to two-step projections LD + -→ C(B + ∞ , X) -→ B + . We shall describe two such examples. 8.1. Action of braids on sequences of integers. Braids act on sequences of integers via their permutations. Indeed, the rule (8.1) (x 1 , ..., x n ) • σ i = (x 1 , ..., x i-1 , x i+1 , x i , x i+2 , ..., x n ).

defines an action of B + n on N n , whence a partial action of B + ∞ on N * , where N * denotes the set of all finite sequences in N. In this way, we obtain a new category C(B + ∞ , N * ), which clearly projects to B + . We shall now describe an explicit projection of LD + onto this category. We recall that terms have been defined to be bracketed expressions constructed from a fixed sequence of variables x 1 , x 2 , ... (or as binary trees with leaves labelled with π π

x p x q x p x q (..., p, q, ...) (..., q, p, ...) σ i D 1 i-1 variables x p ), and that, for t a term and α a binary address, t /α denotes the subterm of t whose root, when t is viewed as a binary tree, has address α. where π is defined for ht(t) = n by π(t) = (var R (t /0 ), var R (t /10 ), ..., var R (t /1 n-1 0 )), var R (t) denoting the index of the righmost variable occurring in t.

So, a typical morphism of B + is ((1, 2, 2), σ 1 , (2, 1, 2)), and the projection of terms to sequences of integers is given by x p1

x p2

x pn → (p 1 , p 2 , ..., p n ). π

Proof (Sketch). The point is to check that the action of the LD-law on the indices of the right variables of the subterms with addresses 1 i 0 is compatible with the action of braids on sequences of integers. It suffices to consider the basic case of D 1 i-1 , and the expected relation is shown in Figure 9. Details are easy. Note that, for symmetry reasons, the category B + is not only left-Garside, but even Garside.

8.2. Action of braids on LD-systems. The action of positive braids on sequences of integers defined in (8.1) is just one example of a much more general situation, namely the action of positive braids on sequences of elements of any LDsystem. It is well known-see, for instance, [18, Chapter I]-that, if (S, ⋆) is an LD-system, i.e., ⋆ is a binary operation on S that obeys the LD-law, then (8.2) (x 1 , ..., x n ) • σ i = (x 1 , ..., x i-1 , x i ⋆ x i+1 , x i , x i+2 , ..., x n ) induces a well defined action of the monoid B + n on the set S n , and, from there, a partial action of B + ∞ on the set S * of all finite sequences of elements of S.

Proposition 8.2. Assume that (S, ⋆) is an LD-system, and let B + S be the category associated with the partial action (8.2) of B + ∞ on S * . Then B + S is a left-Garside category, and, for all s 1 , s 2 , ... in S, the projection [π, ht] of LD + onto B + factors through B + S into LD + [π, π S ] ------→ B +

S

[id, lg] ------→ B + , where π S is defined for ht(t) = n by π S (t) = (eval S (t /0 ), ..., eval S (t /1 n-1 0 )), eval S (t) being the evaluation of t in (S, ⋆) when x p is given the value s p for each p.

We skip the proof, which is an easy verification similar to that of Proposition 8.1. When (S, ⋆) is N equipped with x ⋆ y = y and we map x p to p, we obtain the category B + of Proposition 8.1. In this case, the (partial) action of braids is not constant as in the case of B + , but it factors through an action of the associated permutations, and it is therefore far from being free. By contrast, if we take for S the braid group B ∞ with ⋆ defined by x ⋆ y = x sh(y) σ 1 sh(x) -1 , where we recall sh is the shift endomorphism of B ∞ that maps σ i to σ i+1 for each i, and if we send x p to 1 (or to any other fixed braid) for each p, then the corresponding action (8.2) of B + ∞ on (B ∞ ) * is free, in the sense that a = a ′ holds whenever s • a = s • a ′ holds for at least one sequence s in (B ∞ ) * : this follows from Lemma III.1.10 of [START_REF] Dehornoy | Ordering Braids[END_REF]. This suggests that the associated category C(B + ∞ , (B ∞ ) * )) has a very rich structure.

Appendix: Other algebraic laws

The above approach of self-distributivity can be developed for other algebraic laws as well. However, at least from the viewpoint of Garside structures, the case of self-distributivity seems quite particular.

The case of associativity. Associativity is the law x(yz) = (xy)z. It is syntactically close to self-distributivity, the only difference being that the variable x is not duplicated in the right hand side. Let us say that a term t ′ is an A-expansion of another term t if t ′ can be obtained from t by applying the associativity law in the left-to-right direction only, i.e., by iteratively replacing subterms of the form t 1 ⋆ (t 2 ⋆ t 3 ) by the corresponding term (t 1 ⋆ t 2 ) ⋆ t 3 . Then the counterpart of Proposition 4.5 is true, i.e., two terms t, t ′ are equivalent up to associativity if and only if they admit a common A-expansio, a trivial result since every size n term t admits as an A-expansion the term φ(t) obtained from t by pushing all brackets to the left.

As in Sections 4.3 and 5.2, we can introduce the category A + 0 whose oblects are terms, and whose morphisms are pairs (t, t ′ ) with t ′ an A-expansion of t. As in Section 5.1, we can take positions into account, using A α when associativity is applied at address α, and introduce a monoid A + that describes the connections between the generators A α [START_REF] Dehornoy | Geometric presentations of Thompson's groups[END_REF]. Here the relations of Lemma 5.3 are to be replaced by analogous new relations, among which the MacLane-Stasheff Pentagon relations A 2 α = A α1 A α A α0 . The monoid A + turns out to be a well known object: indeed, it is (isomorphic to) the submonoid F + of R. Thompson's group F generated by the standard generators x 1 , x 2 , ... [START_REF] Cannon | Introductory notes on Richard Thompson's groups[END_REF].

Finally, as in Section 5, we can introduce the category A + , whose objects are terms, and whose morphisms are triples (t, a, t ′ ) with a in A + and t • a = t ′ . Using ψ(t) for the term obtained from t by pushing all brackets to the right, we have Proposition. The categories A + 0 and A + are isomorphic; A + 0 is left-Garside with Garside map t → (t, φ(t)), and right-Garside with Garside map t → (ψ(t), t). This result might appear promising. It is not! Indeed, the involved Garside structure(s) is trivial: the maps φ and ψ are constant on each orbit of the action of A + on terms, and it easily follows that every morphism in A + 0 and A + is left-simple and right-simple so that, for instance, the greedy normal form of any morphism always has length one 3 . The only observation worth noting is that A + provides an example where the left-and the right-Garside structures are not compatible, and, therefore, we have no Garside structure in the sense of Definition 1.6.

Central duplication. We conclude with still another example, namely the exotic central duplication law x(yz) = (xy)(yz) of [START_REF] Dehornoy | Study of an identity[END_REF]. The situation there turns out to be very similar to that of self-distributivity, and a nontrivial left-Garside structure appears. As there is no known between this law and other widely investigated objects like braids, it is probably not necessary to go into details.

1 . 1 . 3 .

 113 -divisibility is defined symmetrically: a right-divides b if b = ca holds for some c. Then the notion of a right-(pre)Garside monoid is defined by replacing leftdivisibility by right-divisibility and left-product by right-product in Definition 1.Definition We say that a monoid M is Garside with Garside element ∆ if M is both left-Garside with left-Garside element ∆ and right-Garside with right-Garside element ∆.

2. 1 .

 1 Simple morphisms and the functor φ.

Definition

Figure 1 .

 1 Figure 1. Relation (2.1): the Garside map ∆ viewed as a natural transformation from the identity functor to the functor φ.

  • a and x • b are defined. By Proposition 2.11, (x, a, x • a) and (x, b, x • b) admit a right-lcm (x, c, z) in the category C(M, X). By construction, c is a common right-multiple of a and b in M . As M is assumed to satisfy (LG 2 ), a and b admit a right-lcm in M . (ii) Fix now x in X, and let a, b belong to M x , i.e., assume that x • a and x • b are defined. Then (x, a, x • a) and (x, b, x • b) are morphisms of C(M, X). As above, they admit a right-lcm, which must be (x, c, x • c) when c is the right-lcm of a and b. Hence c belongs to M x .

Figure 3 .

 3 Figure 3. Adding one simple factor g d on the right of a simple sequence (f1, ..., f d ): compute the normal form(g d-1 f ′ d ) of f d g d ), then the normal form (g d-2 f ′ d-2 ) of f d-1 g d-1, and so on from right to left; the sequence (g0, f ′ 1 , ..., f ′ d ) is normal.

Proposition 4 . 5 .

 45 [START_REF] Dehornoy | Free distributive groupoids[END_REF] Two terms are LD-equivalent if and only if they admit a common LD-expansion.

Figure 4 .

 4 Figure 4. The fundamental LD-expansion φ(t) of a term t, recursive definition: φ(t1 ⋆ t2) is obtained by distributing φ(t1) everywhere in φ(t2).

Lemma

Lemma 5 . 2 .

 52 Definition 5.1 gives a partial action of the free monoid {D α | α ∈ A} * on T (the set of terms), in the sense of Definition 1.7.

Figure 7 .Lemma 5 . 3 .

 753 Figure 7. Relations between Dα-expansions: the critical case. We read that the action of DǫD1Dǫ and D1DǫD1D0 coincide.

Definition 5 . 4 .Lemma 5 .3 immediately implies Proposition 5 . 5 .

 54555 Let R LD be the family of all relations of Lemma 5.3. We define LD + to be the monoid{D α | α ∈ A} | R LD + .The partial action of the free monoid {D α | α ∈ A} * on terms induces a well defined partial action of the monoid LD + .

Definition 6 . 5 .

 65 Let S be any set. A map C : S × S → S * is called a complement on S. Then, we denote by R C the family of all relations aC(a, b) = bC(b, a) with a = b in S, and by C the unique (possibly partial) map of S * × S * to S * that extends C and obeys the recursive rules (6.1)

Proposition 6 . 6 (

 66 [START_REF] Dehornoy | Complete positive group presentations[END_REF] or[18, Prop. II.2.5.]). Assume that M is a monoid satisfying (LG 0 ) and admitting the presentation S, R C + , where C is a complement on S. Then the following are equivalent:(i) The monoid M is left-preGarside;(ii) For all a, b, c in S, we have (6.2) C( C( C(a, b), C(a, c)), C( C(b, a), C(b, c))) = ǫ.

Example 6 . 14 .

 614 Assume a = D ǫ , b = D 1 , and t = x⋆ (x⋆ (x⋆ x)). Then t•a and t•b are defined. On the other hand, we have φ

Figure 8 .

 8 Figure 8. The left diagram shows an instance of Relation (6.4): for the considered choice of t, we find ∆t = DǫD1Dǫ, ∆t•D 1 = D1DǫD0D00D1D10, leading to φt(D1) = Dǫ and φt(Dǫ) = D0D1. Here φt(Dǫ) and φt(D1) are left-coprime, so (6.4) is true. The right diagram shows that the counterpart involving lcm's fails.

4 )

 4 holds for t = (x⋆(x⋆x))⋆(x⋆(x⋆x)) with a = D 0 and b = D 1 ; the values are φ t (D 0 ) = D 000 D 010 D 100 D 110 and φ t (D 1 ) = D ǫ .Proposition 6.13 leads to a realistic program that would reduce the proof of the Embedding Conjecture to a (long) sequence of verifications. Indeed, it is shown in Proposition VIII.5.15 of[START_REF] Dehornoy | Braids and Self-Distributivity[END_REF] that every simple element a of LD + admits a unique expression of the form

(7. 1 )

 1 For all a, b in S, we have C(π(a), π(b)) = π(C(a, b)).

  Let a, b, c be elements of S. By hypothesis, there exist a, b, c in S satisfying π(a) = a, π(b) = b, π(c) = c. As M is left-preGarside, by the direct implication of Proposition 6.6, the relation (6.2) involving a, b, c is true in S * . Applying π and using (7.2), we deduce that the relation (6.2) involving a, b, c is true in S * . Then, as M satisfies (LG 0 ) by hypothesis, the converse implication of Proposition 6.6 implies that M is left-preGarside. Then, by definition, the relations aC(a, b) = bC(b, a) with a, b ∈ S make a presentation of M . Now, for a, b in S, we find π(a)C(π(a), π(b)) = π(aC(a, b)) = π(bC(b, a)) = π(b)C(π(b), π(a))

7. 2 .Proposition 7 . 3 .

 273 The case of LD + and B + . Applying the criterion of Section 7.1 to the categories LD + and B + is easy. The monoid B +

Figure 9 .

 9 Figure 9. Compatibility of the action of D 1 i-1 on sequences of "sub- right" variables and of the action of σ i on sequences of integers.

Proposition 8 . 1 .

 81 Let B + be the category associated with the partial action (8.1) of B + ∞ on N * . Then B + is a Garside category, and the projection [π, ht] of LD + onto B + factors through B + intoLD + [π, π] ----→ B + [id, lg] -----→ B + ,

  4.7. [17] [18, Lemmas V.3.11 and V.3.12] (i) The term φ(t) is an LDexpansion of each atomic LD-expansion of t. (ii) If t ′ is an LD-expansion of t, then φ(t ′ ) is an LD-expansion of φ(t). Sketch of proof. One uses induction on the size of the involved terms. Once Lemma 4.7 is established, an easy induction on d shows that, if there exists a length d sequence of atomic LD-expansions connecting t to t ′ , then φ The category LD + 0 . A category (and a quiver) is naturally associated with every graph, and the previous results invite to introduce the category associated with the LD-expansion relation → LD . Definition 4.8. We denote by LD + 0 the category whose objects are terms, and whose morphisms are pairs of terms (t, t ′ ) satisfying t → LD t ′ .

d (t) is an LD-expansion of t ′ . Then a final induction on the length of a zigzag connecting t to t ′ shows that, if t and t ′ are LD-equivalent, then φ d (t) is an LD-expansion of t ′ for sufficiently large d (namely for d at least the number of "zag"s in the zigzag).

4.3.

In[START_REF] Dehornoy | Free distributive groupoids[END_REF] and[START_REF] Dehornoy | Braids and Self-Distributivity[END_REF], ∂ is used instead of φ, an inappropriate notation in the current context.

Proof. Assume that f, g are morphisms of C that satisfy φ(f ) = φ(g). As φ is a functor, we first deduce φ(∂ 0 f ) = ∂ 0 (φ(f )) = ∂ 0 (φ(g)) = φ(∂ 0 g), hence ∂ 0 f = ∂ 0 g as φ is injective on objects. A similar argument gives ∂ 1 f = ∂ 1 g. Then, (2.1) gives

If we can cancel ∆(∂ 1 f ) on the right, we deduce f = g and, therefore, (i) implies (ii).

Conversely, assume that h is simple and f h = gh holds. By multiplying by h * , we deduce f hh * = ghh * , i.e., f ∆(∂ 0 h) = g∆(∂ 0 h). As we have

hence φ(f ) = φ(g) by left-cancelling ∆(∂ 0 f ). If (ii) holds, we deduce f = g, i.e., h is right-cancellable. As simple morphisms generate Hom(C), we deduce that every morphism is right-cancellable and, therefore, (ii) implies (i).

It is clear that (ii) implies (iii). So assume that C is regular and (iii) holds. Let f, g satisfy φ(f ) = φ(g). Let (f 1 , ..., f d ) and (g 1 , ..., g e ) be the normal forms of f and g, respectively. The regularity assumption implies that every length 2 subsequence of (φ(f 1 ), ..., φ(f d )) and (φ(g 1 ), ..., φ(g e )) is normal. Moreover, (iii) guarantees that φ(f i ) and φ(g j ) is nontrivial. Hence (φ(f 1 ), ..., φ(f d )) and (φ(g 1 ), ..., φ(g e )) are normal. As φ is a functor, we have φ(f 1 )...φ(f d ) = φ(f ) = φ(g) = φ(g 1 )...φ(g e ), and the uniqueness of the normal form implies d = e, and φ(f i ) = φ(g i ) for each i. Then (iii) implies f i = g i for each i, hence f = g. So, in order to prove Theorem 6.2, it suffices to show that the category LD + satisfies the hypotheses of Proposition 6.8, and this is what we do now. Lemma 6.9. The functor φ of LD + is injective on objects, i.e., on terms.

Proof. We show using induction on the size of t that φ(t) determines t. The result is obvious if t has size 0. Assume t = t 0 ⋆ t 1 . By construction, the term φ(t) is obtained by substituting every variable x i occurring in the term φ(t 1 ) with the term φ(t 0 ) ⋆ x i . Hence φ(t 0 ) is the 1 n-1 0th subterm of φ(t), where n is the common right-height of t and φ(t). From there, φ(t 1 ) can be recovered by replacing the subterms φ(t 0 ) ⋆ x i of φ(t) by x i . Then, by induction hypothesis, t 0 and t 1 , hence t, can be recovered from φ(t 1 ) and φ(t 0 ). Lemma 6.10. The functor φ of LD + is injective on simple morphisms.

Proof. Assume that f, f ′ are morphisms of LD + satisfying φ(f ) = φ(f ′ ), say f = (t, a, s) and f ′ = (t ′ , a ′ , s ′ ). The explicit description of Lemma 2.7 implies φ(t) = φ(t ′ ), hence t = t ′ by Lemma 6.9. Similarly, we have φ(s) = φ(s ′ ), hence s ′ = s. Therefore, we have t • a = t • a ′ = s. By Proposition VII.126 of [START_REF] Dehornoy | Braids and Self-Distributivity[END_REF], we deduce that t • a = t • a ′ holds for every term t for which both t • a and t • a ′ are defined. Then Proposition IX.6.6 of [START_REF] Dehornoy | Braids and Self-Distributivity[END_REF] implies a = a ′ provided a or a ′ is simple.

We can now complete the argument.

Proof of Theorem 6.2. The category LD + is left-Garside, with an associated functor φ that is injective both on objects and on simple morphisms. By Proposition 6.8,