aa r X i v : . [ m a t h . G R ] O c t LEFT-GARSIDE CATEGORIES, SELF-DISTRIBUTIVITY, ANDBRAIDS
PATRICK DEHORNOY
Abstract.
In connection with the emerging theory of Garside categories, wedevelop the notions of a left-Garside category and of a locally left-Garsidemonoid. In this framework, the connection between the self-distributivitylaw LD and braids amounts to the result that a certain category associatedwith LD is a left-Garside category, which projects onto the standard Garsidecategory of braids. This approach leads to a realistic program for establish-ing the Embedding Conjecture of [Dehornoy, Braids and Self-distributivity,Birkha¨user (2000), Chap. IX].
The notion of a Garside monoid emerged at the end of the 1990’s [24, 19] as adevelopment of Garside’s theory of braids [32], 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 [4],Digne–Michel [27], and Krammer [38] introduced the notion of a Garside categoryas a further extension, and they used it to capture new, nontrivial examples andimprove our understanding of their algebraic structure. The concept of a Garsidecategory is also implicit in [25] and [35], and maybe in the many diagrams of [18].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 structuresthat obey it, in particular in set theory [16, 40] and in low-dimensional topology [36,30, 44]. A rather extensive theory was developed in the decade 1985-95 [18].Investigating self-distributivity in the light of Garside categories seems to be agood idea. It turns out that a large part of the theory developed so far can besummarized 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 opportu-nity to restate a number of previously unrelated properties in a new language thatis natural and should make them more easily understandable—this is probably notuseless. In particular, the connection between self-distributivity and braids is nowexpressed in the simple statement: There exists a right-lcm preserving surjective functor of LD + to theGarside category of positive braids, (second part of Theorem 6.1). This result allows one to recover most of the usualalgebraic properties of braids as a direct application of the properties of LD + : Mathematics Subject Classification.
Key words and phrases.
Garside category, Garside monoid, self-distributivity, braid, greedynormal form, least common multiple, LD-expansion. 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 estab-lishing the Embedding Conjecture. The latter is the most puzzling open questioninvolving free self-distributive systems. Among others, it says that the equivalenceclass of any bracketed expression under self-distributivity is a semilattice, i.e. , anytwo expressions admit a least upper bound with respect to a certain partial order-ing. Many equivalent forms of the conjecture are know [18, Chapter IX]. At themoment, no complete proof is known, but we establish the following new result
Unless the left-Garside category LD + is not regular, the EmbeddingConjecture 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 cate-gory, quite different from all previously known examples of Garside categories. Inparticular, being strongly asymmetric, LD + is not a Garside category. The interestof investigating such objects per se is not obvious, but the existence of a nontrivialexample such as LD + seems to be a good reason, and a help for orientating furtherreaserch. In particular, our approach emphasizes the role of locally left-Garsidemonoids : 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 roleof the Garside element and are indexed by a set on which the monoid M partiallyacts. Most of the properties of left-Garside monoids extend to locally left-Garsidemonoids, 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 [27](up to a slight change in the formal setting, see Remark 2.6). Several proofs inSection 2 and 3 use arguments that are already present, in one form or another,in [1, 48, 28, 29, 12, 19, 35] and now belong to folklore. Most appear in the unpub-lished paper [27] by Digne and Michel, and are implicit in Krammer’s paper [38].Our reasons for including such arguments here is that adapting them to the currentweak context requires some polishing, and that it makes it natural to introduce ourtwo main new notions, namely locally Garside monoids and regular left-Garsidecategories.The paper is organized in two parts. The first one (Sections 1 to 3) containsthose general results about left-Garside categories and locally left-Garside monoidsthat will be needed in the sequel, in particular the construction and properties ofthe greedy normal form. The second part (Sections 4 to 8) deals with the specificcase of the category LD + and its connection with braids. Sections 4 and 5 reviewbasic facts about the self-distributivity law and explain the construction of thecategory LD + . Section 6 is devoted to proving that LD + is a left-Garside categoryand to showing how the results of Section 3 might lead to a proof of the EmbeddingConjecture. In Section 7, we show how to recover the classical algebraic propertiesof braids from those of LD + . Finally, we explain in Section 8 some alternative This is not the notion of a locally Garside monoid in the sense of [27]; we think that thename “preGarside” is more relevant for that notion, which involves no counterpart of any Garsideelement or map, but is only the common substratum of all Garside structures.
EFT-GARSIDE CATEGORIES, SELF-DISTRIBUTIVITY, AND BRAIDS 3 solutions for projecting LD + to braids. In an appendix, we briefly describe whathappens when the associativity law replaces the self-distributivity law: here also aleft-Garside category appears, but a trivial one.We use N for the set of all positive integers.1. Left-Garside categories
We define left-Garside categories and describe a uniform way of constructingsuch categories from so-called locally left-Garside monoids, which are monoids witha convenient partial action.1.1.
Left-Garside monoids.
Let us start from the now classical notion of a Gar-side monoid. Essentially, a Garside monoid is a monoid in which divisibility hasgood properties, and, in addition, there exists a distinguished element ∆ whosedivisors encode the whole structure. Slightly different versions have been consid-ered [24, 19, 26], the one stated below now being the most frequently used. In thispaper, we are interested in one-sided versions involving left-divisibility only, hencewe 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 thereexists 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 agreatest lower bound c with respect to , the latter is called a greatest commonleft-divisor , or left-gcd , of a and b , denoted c = gcd( a, b ). Similarly, a -least upperbound 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 left-generates S ′ if every nontrivial element of S ′ admits at least one nontrivial left-divisor belonging to S . Definition 1.1.
We say that a monoid M is left-preGarside if( LG ) for each a in M , every ≺ -increasing sequence in Div( a ) is finite,( LG ) M is left-cancellative,( LG ) M admits local right-lcm’s.An element ∆ of M is called a left-Garside element if( LG ) Div(∆) left-generates M , and a ∆ implies ∆ a ∆.We say that M is left-Garside if it is left-preGarside and possesses at least oneleft-Garside element.Using “generates” instead of “left-generates” in ( LG ) would make no difference,by the following trivial remark—but the assumption ( LG ) is crucial, of course. Lemma 1.2.
Assume that M is a monoid satisfying ( LG ) . Then every subset S left-generating M generates M .Proof. Let a be a nontrivial element of M . By definition there exist a = 1 in S and a ′ satisfying a = a a ′ . If a ′ is trivial, we are done. Otherwise, there exist a = 1 PATRICK DEHORNOY in S and a ′′ satisfying a ′ = a a ′′ , and so on. The sequence 1, a , a a , ... is ≺ -increasing and it lies in Div( a ), hence it must be finite, yielding a = a ...a d with a , ..., a d in S . (cid:3) Right-divisibility is defined symmetrically: a right-divides b if b = ca holds forsome c . Then the notion of a right-(pre)Garside monoid is defined by replacing left-divisibility by right-divisibility and left-product by right-product in Definition 1.1. Definition 1.3.
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 ∆.The equivalence of the above definition with that of [26] is easily checked. Theseminal example of a Garside monoid is the braid monoid B + n equipped with Gar-side’s fundamental braid ∆ n , see for instance [32, 29]. Other classical examples arefree abelian monoids and, more generally, all spherical Artin–Tits monoids [10], aswell as the so-called dual Artin–Tits monoids [9, 4]. Every Garside monoid embedsin a group of fractions, which is then called a Garside group .Let us mention that, if a monoid M is left-Garside, then mild conditions implythat it is Garside: essentially, it is sufficient that M is right-cancellative and thatthe left- and right-divisors of the left-Garside element ∆ coincide [19].1.2. Left-Garside categories.
Recently, it appeared that a number of resultsinvolving Garside monoids still make sense in a wider context where categoriesreplace monoids [4, 27, 38]. A category is similar to a monoid, but the productof two elements is defined only when the target of the first is the source of thesecond. In the case of Garside monoids, the main benefit of considering categoriesis that it allows for relaxing the existence of the global Garside element ∆ into aweaker, local version depending on the objects of the category, namely a map fromthe objects to the morphisms.We refer to [43] for some basic vocabulary about categories—we use very littleof it here.
Convention.
Throughout the paper, composition of morphisms is denoted by amultiplication on the right: f g means “ f then g ”. If f is a morphism, the sourceof f is denoted ∂ f , and its target is denoted ∂ f . In all examples, we shall makethe source and target explicit: morphisms are triples ( x, f, y ) satisfying ∂ ( x, f, y ) = x, ∂ ( x, f, y ) = y. A morphism f is said to be nontrivial if f = 1 ∂ f holds.We extend to categories the terminology of divisibility. So, we say that a mor-phism 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 ∂ f = ∂ g . We denote by Div( f ) the collection ofall left-divisors of f .The following definition is equivalent to Definition 2.10 of [27] by F. Digne andJ. Michel—see Remark 2.6 below. Definition 1.4.
We say that a category C is left-preGarside if( LG ) for each f in H om ( C ), every ≺ -ascending sequence in Div( f ) is finite,( LG ) H om ( C ) admits left-cancellation, EFT-GARSIDE CATEGORIES, SELF-DISTRIBUTIVITY, AND BRAIDS 5 ( LG ) H om ( C ) admits local right-lcm’s.A map ∆ : O bj ( C ) → H om ( C ) is called a left-Garside map if, for each object x , wehave ∂ ∆( x ) = x and( LG ) ∆( x ) left-generates Hom( x, − ), and f ∆( x ) implies ∆( x ) f ∆( ∂ f ).We say that C is left-Garside if it is left-preGarside and possesses at least oneleft-Garside map. Example 1.5.
Assume that M is a left-Garside monoid with left-Garside ele-ment ∆. One trivially obtains a left-Garside category C ( M ) by putting O bj ( C ( M )) = { } , H om ( C ( M )) = { } × M × { } , ∆(1) = ∆ . Another left-Garside category b C M can be attached with M , namely taking O bj ( b C ( M )) = M, H om ( b C ( M )) = { ( a, b, c ) | ab = c } , ∆( a ) = ∆ . It is natural to call b C ( M ) the Cayley category of M since its graph is the Cayleygraph of M (defined provided M is right-cancellative).The notion of a right-Garside category can be defined symmetrically, exchang-ing left and right everywhere and exchanging the roles of source and target. Inparticular, the map ∆ and Axiom ( LG ) is to be replaced by a map ∇ satisfying ∂ ∇ ( x ) = x and, using b e < a for “ a right-divides b ”,( LG ) ∇ ( y ) right-generates Hom( − , y ), and ∇ ( y ) e < f implies ∇ ( ∂ y ) f e < ∇ ( y ).Then comes the natural two-sided version of a Garside category [4, 27]. Definition 1.6.
We say that a category C is Garside with Garside map ∆ if C isleft-Garside with left-Garside map ∆ and right-Garside with right-Garside map ∇ satisfying ∆( x ) = ∇ ( ∂ (∆( x )) and ∇ ( y ) = ∆( ∂ ( ∇ ( y )) for all objects x, y .It is easily seen that, if M is a Garside monoid, then the categories C ( M )and b 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 Defi-nition 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 con-structing a left-Garside category starting from a monoid equipped with a partialaction on a set. The trivial examples of Example 1.5 enter this family, and so dothe 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 • x holds for each x in X ,( ii ) ( x • a ) • b = x • ab holds for all x, a, b , this meaning that either both termsare 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 definedfor 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 imply1 ∈ M x , ab ∈ M x ⇔ ( a ∈ M x & b ∈ M x • a ) , ∀ finite S ∃ x ( S ⊆ M x ) . PATRICK DEHORNOY
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 braidmonoid B + n . We recall that B + n is defined for n ∞ by the monoid presentation(1.2) B + n = (cid:28) σ , ... , σ n − (cid:12)(cid:12)(cid:12)(cid:12) σ i σ j = σ j σ i for | i − j | > σ i σ j σ i = σ j σ i σ j for | i − j | = 1 (cid:29) + . 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 O bj ( C ( M, X )) = X, H om ( 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 O bj ( B + ) = N , H om ( B + ) = { ( n, a, n ) | n ∈ B + n } . Define ∆ : O bj ( B + ) → H om ( B + ) by ∆( n ) = ( n, ∆ n , n ). Then the well known factthat B + n is a Garside monoid for each n [32, 37] easily implies that B + is a Garsidecategory (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. Themonoid B + ∞ is not a (left)-Garside monoid, because it is of infinite type and therecannot exist a global Garside element ∆. However, the partial action of (1.3) enableus to restrict to subsets B + n (submonoids in the current case) for which Garsideelements exist: with the notation of (1.1), B + n is ( B + ∞ ) n . Thus the categoricalcontext allows to capture the fact that B + ∞ is, in some sense, locally Garside. It iseasy 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 . Asequence (∆ x ) x ∈ X of elements of M is called a left-Garside sequence for α if, foreach x in X , the element x • ∆ x is defined and( LG ℓoc ) 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-Gars-ide sequence (∆ x ) x ∈ X . Then C ( M, X ) is a left-Garside category with left-Garsidemap ∆ 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 ) 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 ), and, therefore, C ( M, X ) satisfies ( LG ). EFT-GARSIDE CATEGORIES, SELF-DISTRIBUTIVITY, AND BRAIDS 7
Assume ( x, a, y )( y, b ′ , x ′ ) = ( x, b, z )( z, a ′ , x ′ ) in H om ( C ( M, X )). Then ab ′ = ba ′ holds in M . By ( LG ), 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 obviousto check that ( x, c, x • c ) is a right-lcm of ( x, a, y ) and ( x, b, z ) in H om ( C ( M, X )).Hence C ( M, X ) satisfies ( LG ).Assume that ( x, a, y ) is a nontrivial morphism of H om ( C ( M, X )). This meansthat a is nontrivial, so, by ( LG ℓoc ), 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 H om ( C ( M, X )). This implies a ∆ x in M .Then ( LG ℓoc ) in M implies ∆ x a ∆ y . By hypothesis, y • ∆ y is defined, and wehave ( x, a, y )∆( y ) = ( x, a ∆ y , x • a ∆ y ), of which ( x, ∆ x , x • ∆ x ) is a left-divisorin H om ( C ( M, X )). So ( LG ) is satisfied in C ( M, X ). (cid:3) 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-Garsidemonoid. 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 ) reduces to ( LG ). In this case, each ele-ment ∆ x is a left-Garside element in M , and M is a left-Garside monoid. A similarresult 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-Gars-ide 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 • M x contains 1, and it is asubmonoid of M . We show that M x satisfies ( LG ) , ( LG ) , ( LG ), and ( LG ). First, acounter-example to ( LG ) in M x would be a counter-example to ( LG ) in M , hence M x satisfies ( LG ). Similarly, an equality ab = ab ′ with b = b ′ in M x would alsocontradict ( LG ) in M , so M x satisfies ( LG ). Now, assume that a and b admitin 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 partialaction, 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 ), and it is left-preGarside.Next, ∆ x is a left-Garside element in M x . Indeed, let a be any nontrivial elementof M x . By ( LG ℓoc ), there exists a nontrivial divisor a ′ of a satisfying a ′ ∆ x . Bydefinition of a partial action, x • a ′ is defined, so it belongs to M x , and ∆ x left-generates 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 ), i.e. , ∆ x a ∆ x since we assumed that the sequence ∆is constant on M x . So ∆ x a left-Garside in M x . (cid:3) Simple morphisms
We return to general left-Garside categories and establish a few basic results. Asin the case of Garside monoids, an important role is played by the divisors of ∆, alocal notion here.2.1.
Simple morphisms and the functor φ .Definition 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 ∆( ∂ f ). In this case, we denote by f ∗ PATRICK DEHORNOY the unique simple morphism satisfying f f ∗ = ∆( ∂ f ). The family of all simplemorphisms in C is denoted by H om sp ( C ).By definition, every identity morphism 1 x is a left-divisor of every morphismwith 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 ) = ∂ (∆( x ))for x in O bj ( C ), and φ ( f ) = f ∗∗ for f in H om sp ( C ).Although straightforward, the following result is fundamental—and it is the mainargument for stating ( LG ) 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 ), we have f f ∗ = ∆( ∂ f ) f ∆( ∂ f ), hence f ∗ ∆( ∂ f ) byleft-cancelling f . This shows that f ∗ is simple. Applying the result to f ∗ showsthat 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 ). Then g is simple by ( i ) (cid:3) Lemma 2.4.
Assume that C is a left-Garside category. ( i ) The morphisms 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 ∆( ∂ f ) = ∆( ∂ f ) φ ( f ) . Proof. ( i ) Assume f g = 1 x with f = 1 x and g = 1 ∂ f . Then we have1 x ≺ f ≺ f g ≺ f ≺ f g ≺ ..., an infinite ≺ -increasing sequence in Div(1 x ) that contradicts ( LG ).( ii ) Let f be a morphism of C , and let x = ∂ f . If f is trivial, then it is simple, asobserved above. We wish to prove that simple morphisms generate H om ( C ). Owingto Lemma 1.2, it is enough to prove that simple morphisms left-generate H om ( C ), i.e. , that every nontrivial morphism with source x is left-divisible by a simplemorphism with source x , in other words by a left-divisor of ∆( x ). This is exactlywhat the first part of Condition ( LG ) claims.( iii ) Up to now, φ has been defined on objects, and on simple morphisms. Notethat, by construction, (2.1) is satisfied for each simple morphism f . Indeed, apply-ing Definition 2.1 for f and f ∗ gives the relations f f ∗ = ∆( ∂ f ) and f ∗ f ∗∗ = ∆( ∂ f ∗ ) = ∆( ∂ f ) , whence f ∆( ∂ f ) = f f ∗ f ∗∗ = ∆( ∂ f ) f ∗∗ = ∆( ∂ f ) φ ( f ) . Applying this to f = 1 x gives ∆( x ) = ∆( x ) φ (1 x ), hence φ (1 x ) = 1 φ ( x ) by ( LG ).Let f be an arbitrary morphism of C , and let f ...f p and g ...g q be two decom-positions of f as a product of simple morphisms, which exist by ( ii ). Repeatedly EFT-GARSIDE CATEGORIES, SELF-DISTRIBUTIVITY, AND BRAIDS 9 applying (2.1) to f p , ..., f and g q , ..., g gives f ∆( ∂ f ) = f ...f p ∆( ∂ f ) = ∆( ∂ f ) φ ( f ) ...φ ( f p )= g ...g q ∆( ∂ f ) = ∆( ∂ f ) φ ( g ) ...φ ( g q ) . By ( LG ), we deduce φ ( f ) ...φ ( f p ) = φ ( g ) ...φ ( g q ), and therefore there is no ambi-guity in defining φ ( f ) to be the common value. In this way, φ is extended to allmorphisms 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 newis needed here. See Figure 1 for an illustration. (cid:3) ∆( x ) ∆( y ) φ ( x ) φ ( y ) φ ( f ) f ∗ yx f Figure 1.
Relation (2.1) : the Garside map ∆ viewed as a natural trans-formation from the identity functor to the functor φ . Lemma 2.5.
Assume that C is a left-Garside category. Then, for each object x and each simple morphism f , we have (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 = ∂ f . Then we have f f ∗ = ∆( x ), and ∂ ( φ ( f )) = φ ( x ).Applying φ and the above relation, we find φ ( f ) φ ( f ∗ ) = φ (∆( x )) = ∆( φ ( x )) = ∆( ∂ ( φ ( f ))) = φ ( f ) φ ( f ) ∗ . Left-cancelling φ ( f ) yields φ ( f ∗ ) = φ ( f ) ∗ . (cid:3) Remark 2.6.
We can now see that Definition 1.4 is equivalent to Definition 2.10of [27]: 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-Garsidecategory in the sense of [27]. 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 ), implies φ ( f ) = f ∗∗ . So every left-Garside categoryin the sense of [27] 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 partic-ular 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-Garsidesequence (∆ 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 , a , x ),..., , ( x d − , a d , x d ). This implies a = a ...a d in M . The hypothesis that each mor-phism ( x i − , a i , x i ) is simple implies ∆ x i − a i ∆ x i for each i , whence∆ x a ∆ x a a ∆ x ... a ...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 )). (cid:3) Greatest common divisors.
We observe—or rather recall—that left-gcd’salways exist in a left-preGarside category. We begin with a standard consequenceof the noetherianity assumption ( LG ). Lemma 2.8.
Assume that C is a left-preGarside category and S is a subset of H om ( C ) that contains the identity-morphisms and is closed under right-lcm. Thenevery morphism has a unique maximal left-divisor that lies in S .Proof. Let f be an arbitrary morphism. Starting from f = 1 ∂ f , which belongsto S by hypothesis, we construct a ≺ -increasing sequence f , f , ... 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 ) implies that the construction stops after a finitenumber 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 liein S . By construction, g ′ and g ′′ admit a common right-multiple, namely f , hence,by ( LG ), they admit a right-lcm g . By construction, g is a left-divisor of f , and itbelongs to S since g ′ and g ′′ do. The maximality of g and g ′ implies g ′ = g = g ′′ . (cid:3) Proposition 2.9.
Assume that C is a left-preGarside category. Then any twomorphisms 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 ∂ f ,and it is closed under lcm. A left-gcd of f and g is a maximal left-divisor of f lyingin S . Lemma 2.8 gives the result. (cid:3) Least common multiples.
As for right-lcm, the axioms of left-Garside cat-egories only demand that a right-lcm exists when a common right-multiple does. Anecessary condition for such a common right-multiple to exist is to share a commonsource. 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 ...f d with f , ..., f d simple and x = ∂ f , we have (2.4) f ∆( x ) ∆( φ ( x )) ... ∆( φ d − ( x )) . Proof.
We use induction on d . For d = 1, this is the definition of simplicity. Assume d >
2. Put y = ∂ f . Applying the induction hypothesis to f ...f d , we find f = f ( f ...f d ) f ∆( y ) ∆( φ ( y )) ... ∆( φ d − ( y ))= ∆( x ) ∆( φ ( x )) ... ∆( φ d − ( x )) φ d − ( f ) ∆( x ) ∆( φ ( x )) ... ∆( φ d − ( x )) ∆( φ d − ( x )) . EFT-GARSIDE CATEGORIES, SELF-DISTRIBUTIVITY, AND BRAIDS 11
The second equality comes from applying (2.1) d − φ d − ( f ) is simple with source φ d − ( x ). (cid:3) Proposition 2.11.
Assume that C is a left-Garside category. Then any two mor-phisms 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 − ( x )) is a common right-multiple of f and g . Finally, ( LG ) implies that f and g admit a right-lcm. The uniqueness ofthe latter is guaranteed by Lemma 2.4( i ). (cid:3) In a general context of categories, right-lcm’s are usually called push-outs (wher-eas 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 apartial action gives analogous results for all locally left-Garside monoids.
Corollary 2.12.
Assume that M is a locally left-Garside monoid with respect tosome 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 se-quence (∆ 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 • 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 ). Byconstruction, c is a common right-multiple of a and b in M . As M is assumed tosatisfy ( LG ), 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 . (cid:3) Regular left-Garside categories
The main interest of Garside structures is the existence of a canonical normalforms, the so-called greedy normal form [29]. In this section, we adapt the con-struction of the normal form to the context of left-Garside categories—this wasdone in [27] already—and of locally left-Garside monoids. The point here is thatstudying the computation of the normal form naturally leads to introducing thenotion 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 notto be unique in general, and the first step for constructing a normal form consists inisolating a particular simple morphism that left-divides the considered morphism.It will be useful to develop the construction in a general framework where thedistinguished morphisms need not necessarily be the simple ones. Notation.
We recall that, for f, g in H om ( 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 includedin H om ( C ). We say that S is a seed for C if( i ) S left-generates H om ( 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 left-divisible 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 H om sp ( C ) is a seed for C .Proof. First, H om sp ( C ) left-generates H om ( C ) by Condition ( LG ).Next, assume that f, g are simple morphisms sharing the same source x . ByProposition 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 aleft-divisor of ( f \ g ) h , is simple as well by transitivity of .Finally, H om sp ( C ) is closed under left-divisor by definition. (cid:3) 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 3.1implies 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 H om sp ( C ), it is easy to check, for each f in H om ( C ), the equality(3.2) H H om sp ( C ) ( f ) = gcd( f, ∆( ∂ f ));in this case, we shall simply write H ( f ) for H H om sp ( C ) ( f ).3.2. Normal form.
The following result is an adaptation of a result that is clas-sical in the framework of Garside monoids.
Proposition 3.4.
Assume that C is a left-preGarside category and S is a seedfor C . Then every nontrivial morphism f of C admits a unique decomposition (3.3) f = f ...f d , where f , ..., 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 be the S -head of f . Then f belongs to S , it is nontrivial, and we have f = f f ′ for some unique f ′ . If f ′ is trivial, we are done, otherwise we repeat the argument with f ′ . In this way weobtain a ≺ -increasing sequence 1 ∂ f ≺ f ≺ f f ≺ ... . Condition ( LG ) implies EFT-GARSIDE CATEGORIES, SELF-DISTRIBUTIVITY, AND BRAIDS 13 that the construction stops after a finite number of steps, yielding a decompositionof the form (3.3).As for uniqueness, assume that ( f , ..., f d ) and ( g , ..., g e ) are decomposition of f that satisfy the conditions of the statement. We prove ( f , ..., f d ) = ( g , ..., g e ) usinginduction on min( d, e ). First, d = 0 implies e = 0 by Lemma 2.4( i ). Otherwise, thehypotheses imply f = H S ( f ) = g . Left-cancelling f gives two decompositions( f , ..., f d ) and ( g , ..., g e ) of f ...f d , and we apply the induction hypothesis. (cid:3) Definition 3.5.
In the context above, the sequence ( f , ..., f d ) is called the S -normal form of f .When S turns out to be the family H om sp ( C ), the S -normal form will be simplycalled the normal form . The interest of the S -normal form lies in that it is easilycharacterized and easily computed. First, one has the following local characteriza-tion of normal sequences. Proposition 3.6.
Assume that C is a left-preGarside category and S is a seedfor C . Then a sequence of morphisms ( f , ..., f d ) is S -normal if and only if eachlength two subsequence ( f i , f i +1 ) is S -normal. This follows from an auxiliary lemma.
Lemma 3.7.
Assume that ( f , f ) is S -normal and g belongs to S . Then g f f f implies g f f .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 ofthe S -normal form of f show that g ′ belongs to S . By hypothesis, we have both g f f f and f f f f , hence lcm( f, g ) = f g ′ f f f whence g ′ f f byleft-cancelling f . As g belongs to S and ( f , f ) is normal, this implies g ′ f , andfinally g f g ′ f f . (cid:3) Proof of Proposition 3.6.
It is enough to consider the case d = 2, from which aneasy induction on d gives the general case. So we assume that ( f , f ) and ( f , f )are S -normal, and aim at proving that ( f , f , f ) is S -normal. The point is to provethat, if g belongs to S , then g f f f implies g f . So assume g f f f .As ( f , f ) is S -normal, Lemma 3.7 implies g f f . As ( f , f ) is S -normal, thisimplies g f . (cid:3) A computation rule.
We establish now a recipe for inductively computingthe S -normal form, namely determining the S -normal form of gf when that of f isknown and g belongs to S . Proposition 3.8.
Assume that C is a left-preGarside category, S is a seed for C ,and ( f , ..., f d ) is the S -normal form of f . Then, for each g in S , the S -normal formof gf is ( f ′ , ..., f ′ d , g d ) , where g = g and ( f ′ i , g i ) is the S -normal form of g i − f i for i increasing from 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 f f ′ f ′ g g g is commutative and ( f , f ) and ( f ′ , g ) are S -normal. Then ( f ′ , f ′ ) is S -normal. f f f ′ f ′ g g g ...... g d − g d f d f ′ d Figure 2.
Adding one S -factor g on the left of an S -normal sequence ( f , ..., f d ) : compute the S -normal form ( f ′ , g ) of g f , then the S -normal form ( f ′ , g ) of g f , and so on from left to right; the sequence ( f ′ , ..., f ′ d , g d ) is S -normal. So assume that h belongs to S and satisfies h f ′ f ′ . Then, a fortiori, we have h f ′ f ′ g = g f f , hence h g f by Lemma 3.7 since ( f , f ) is S -normal.Therefore we have h f ′ g , hence h f ′ since ( f ′ , g ) is S -normal. (cid:3) The results of Proposition 3.6 and 3.8 apply in particular when C is left-Garsideand S is the family of all simple morphisms, in which case they involve the standardnormal form.In the case of lcm’s, Corollary 2.12 shows how a result established for generalleft-Garside categories can induce a similar result for locally left-Garside monoids.The situation is similar with the normal form, provided some additional assumptionis satisfied. Definition 3.9.
A left-Garside sequence (∆ x ) x ∈ X witnessing that a certain monoidis 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 structureof the monoid B + ∞ is coherent. Indeed, a positive n -strand braid a is a left-divisorof ∆ n if and only if it is a left-divisor of ∆ n ′ for every n ′ > n . The reason is thatbeing simple is an intrinsic property of positive braids: a positive braid is simpleif and only if it can be represented by a braid diagram in which any two strandscross at most once [28]. Proposition 3.10.
Assume that M is a locally left-Garside monoid associated witha 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, andthe counterpart of Propositions 3.6 and 3.8 hold for the Σ -normal form in M .Proof. Axiom ( LG ℓoc ) guarantees that every nontrivial element of M is left-divisibleby some nontrivial element of Σ. Then, by hypothesis, 1 x ∆ x holds for eachobject x . Then, assume a, b ∈ Σ. There exists x such that x • a and x • b is defined. Bydefinition 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 , andlcm( a, b ) ∈ Σ. So there exists c satisfying a ( a \ b ) c = ∆ x . By ( LG ℓoc ), 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 isa seed for M in the sense of Definition 3.1.As, by definition, M is a left-preGarside monoid, Proposition 3.4 applies, guar-anteeing the existence and uniqueness of the Σ-normal form on M , and so doPropositions 3.6 and 3.8. (cid:3) EFT-GARSIDE CATEGORIES, SELF-DISTRIBUTIVITY, AND BRAIDS 15 f ′ f g g ...... g d − g d − g d f ′ d − f d − f ′ d f d Figure 3.
Adding one simple factor g d on the right of a simple sequence ( f , ..., f d ) : compute the normal form ( g d − f ′ d ) of f d g d ) , then the normalform ( g d − f ′ d − ) of f d − g d − , and so on from right to left; the sequence ( g , f ′ , ..., f ′ d ) is normal. Thus, the good properties of the greedy normal form are preserved when the as-sumption that a global Garside element ∆ exists is replaced by the weaker assump-tion 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 ofthe recipe of Proposition 3.8 involving right-multiplication by an element of theseed instead of left-multiplication. Such a counterpart exists but, interestingly, thesituation is not symmetric, and we need a new argument. The latter demands thatthe considered category satisfies an additional condition, which is automaticallysatisfied 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 simplemorphisms, and not the case of a general left-preGarside category with an arbitraryseed—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 , f simple with ∂ f = ∂ f ,(3.4) ( f , f ) normal implies ( φ ( f ) , φ ( f )) normal. Proposition 3.12.
Assume that C is a regular left-Garside category, that ( f , ..., f d ) is the normal form of a morphism f , and that g is simple. Then the normal formof f g is ( g , f ′ , ..., f ′ d ) , where g d = g and ( g i − , f ′ i ) is the normal form of f i g i for i decreasing from d to —see Figure 3. We begin with an auxiliary observation.
Lemma 3.13.
Assume that C is a left-Garside category and f , f are simple mor-phisms satisfying ∂ f = ∂ f . Then ( f , f ) is normal if and only if f ∗ and f areleft-coprime, i.e. , gcd( f ∗ , f ) is trivial.Proof. The following equalities always hold: H ( f f ) = gcd( f f , ∆( ∂ f )) = gcd( f f , f f ∗ ) = f gcd( f , f ∗ ) . Hence ( f , f ) is normal, i.e. , f = H ( f f ) holds, if and only if f = f gcd( f , f ∗ )does, which is gcd( f , f ∗ ) = 1 ∂ f as left-cancelling f is allowed. (cid:3) Proof of Proposition 3.12.
As in the case of Proposition 3.8 it is enough to considerthe case d = 2, and therefore it is enough to prove Claim.
Assume that the diagram f ′ f ′ f f g g g is commutative and ( f , f ) and ( g , f ) are normal. Then ( f ′ , f ′ ) is normal. To prove the claim, we introduce the morphisms g ∗ , g ∗ , g ∗ defined by g i g ∗ i = ∆( ∂ g i )(Definition 2.1). Then the diagram f ′ f ′ f f g g g g ∗ g ∗ g ∗ φ ( f ) φ ( f ) is commutative. Indeed,appying (2.1), we find g f ′ g ∗ = f g g ∗ = f ∆( ∂ g ) = f ∆( ∂ f )= ∆( ∂ f ) φ ( f ) = ∆( ∂ g ) φ ( f ) = g g ∗ φ ( f ) , hence f ′ g ∗ = g ∗ φ ( f ) by left-cancelling g . A similar argument gives f ′ g ∗ = g ∗ φ ( f ).Assume that h is simple and satisfies h f ′ f ′ . We deduce h f ′ f ′ g ∗ = g ∗ φ ( f ) φ ( f ) . By hypothesis, ( f , f ) is normal. Hence the hypothesis that C is regular impliesthat ( φ ( f ) , φ ( f )) is normal as well. By Lemma 3.7, h g ∗ φ ( f ) φ ( f ) implies h g ∗ φ ( f ) = f ′ g ∗ . We deduce h gcd( f ′ f ′ , f ′ g ∗ ) = f ′ gcd( f ′ , g ∗ ). ByLemma 3.13, the hypothesis that ( g , f ′ ) is normal implies gcd( g ∗ , f ′ ) = 1, and,finally, we deduce h f ′ , i.e. , ( f ′ , f ′ ) is normal. (cid:3) Remark 3.14.
It might be tempting to mimick the arguments of this section inthe general framework of a left-preGarside category C and a seed S , provided someadditional conditions are satisfied. However, it is unclear that the extension canbe 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 ∂ f only, then the map ∂ f f f ∗ is aleft-Garside map and we are back to left-Garside categories.3.5. Regularity criteria.
We conclude with some sufficient conditions implyingregularity. In particular, we observe that, in the two-sided case, regularity is auto-matically satisfied.
Lemma 3.15.
Assume that C is a left-Garside category C . Then a sufficient con-dition for C to be regular is that the functor φ is bijective on H om ( C ) .Proof. Assume that C is a left-Garside category and φ is bijective on H om ( C ). Firstwe 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 isa 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 thecondition 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 = ∆( ∂ φ ( f )). As φ is surjective, there exists g ′ satisfying g = φ ( g ′ ). Applying (2.2), we obtain φ ( f g ′ ) = ∆( ∂ φ ( f )) = φ (∆( ∂ f ), hence f g ′ = ∆( ∂ f ) by injectivity of φ .Finally, assume that ( f , f ) is normal, and g is a simple morphism left-dividing φ ( f ) φ ( f ), hence satisfying gh = φ ( f ) φ ( f ) for some h . As φ is surjective, wehave 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 ) φ ( f ), hence g ′ h ′ = f f since φ is a functor and it is injective.The hypothesis that ( f , f ) is normal implies g ′ f , hence g = φ ( g ′ ) φ ( f ). So( φ ( f ) , φ ( f )) is normal, and C is regular. (cid:3) EFT-GARSIDE CATEGORIES, SELF-DISTRIBUTIVITY, AND BRAIDS 17
Proposition 3.16.
Every Garside category is regular.Proof.
Assume that C is a left-Garside with respect to ∆ and right-Garside withrespect to ∇ satisfying ∆( x ) = ∇ ( x ′ ) for x ′ = ∂ ∆( x ). Put ψ ( x ′ ) = ∂ ∇ ( x ′ ) for x ′ in O bj ( C ) and, for g simple in H om ( C ), hence a right-divisor of ∇ ( ∂ g ), denoteby ∗ g the unique simple morphism satisfying ∗ gg = ∇ ( ∂ g ), and put ψ ( g ) = ∗∗ g .Then arguments similar to those of Lemma 2.4 give the equality(3.5) ∇ ( ∂ g ) g = ψ ( g ) ∇ ( ∂ 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 . Asymmetric argument gives φ ( ψ ( g )) = g for each g , and we conclude that ψ is theinverse of φ , which is therefore bijective. Then we apply Lemma 3.15. (cid:3) Remark 3.17.
The above proof shows that, if C is a left-Garside category that isGarside, then the associated functor φ is bijective both on O bj ( C ) and on H om ( 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 regularitycriteria that are close to the definition and will be useful in Section 6. We recallthat 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 preservesthe head function on product of two simples: for f , f simple with ∂ f = ∂ f , (3.8) H ( φ ( f f )) = φ ( H ( f f )); Proof.
Assume that C is regular, and let f , f satisfy ∂ f = ∂ f . Let ( f ′ , f ′ ) bethe formal form of f f —which has length 2 at most by Proposition 3.8. Then,( φ ( f ′ ) , φ ( f ′ )) is normal and satisfies φ ( f ′ ) φ ( f ′ ) = φ ( f f ), so ( φ ( f ′ ) , φ ( f ′ )) is thenormal form of f ( f f ). Hence we have H ( f f ) = f ′ and H ( φ ( f f )) = φ ( f ′ ),which is (3.8).Conversely, assume (3.8) and let ( f , f ) be normal. By construction, we have f = H ( f f ), hence φ ( f ) = H ( φ ( f f )) by hypothesis. This means that thenormal form of φ ( f f ) is ( φ ( f ) , g ) for some g satisfying φ ( f f ) = φ ( f ) g . Now φ ( f ) is such a morphism g , and, by ( LG ), it is the only one. So the normal formof φ ( f f ) is ( φ ( f ) , φ ( f )), and C is regular. (cid:3) Proposition 3.19.
Assume that C is a left-Garside category C . Then two sufficientconditions for C to be regular are ( i ) The functor φ preserves left-coprimeness of simple morphisms: for f, g simplewith ∂ f = ∂ 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 ∂ f = ∂ 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 alsogcd( φ ( f ) ∗ , φ ( g )) = 1, which, by Lemma 3.13 again, means that ( φ ( f ) , φ ( g )) isnormal. Hence C is regular.On the other hand, it is clear that ( ii ) implies ( i ). (cid:3) Self-distributivity
We quit general left-Garside categories, and turn to the description of one partic-ular example, namely a certain category (two categories actually) associated withthe left self-distributive law. The latter is the algebraic law(LD) x ( yz ) = ( xy )( xz )extensively investigated in [18].We first review some basic results about this law and the associated free LD-systems, i.e. , the binary systems that obey the LD-law. The key notion is thenotion of an LD-expansion, with two derived categories LD + and LD + that will beour main subject of investigation from now on.4.1. Free LD-systems.
For each algebraic law (or family of algebraic laws), thereexist universal objects in the category of structures that satisfy this law, namelythe free systems. Such structures can be uniformly described as the quotient ofsome absolutely free structures under a convenient congruence.
Definition 4.1.
We let T n be the set of all bracketed expressions involving vari-ables x , ..., x n , i.e. , the closure of { x , ..., x n } under t ⋆ t = ( t )( t ). We use T for the union of all sets T n . Elements of T are called terms .Typical terms are x , x ⋆ x , x ⋆ ( x ⋆ x ), etc. It is convenient to think of termsas rooted binary trees with leaves indexed by the variables: the trees associatedwith the previous terms are • x , x x , and x x x , respectively. The system ( T n , ⋆ )is the absolutely free system (or algebra) generated by x , ..., x n , and every binarysystem generated by n elements is a quotient of this system. So is in particular thefree LD-system of rank n . Definition 4.2.
We denote by = LD the least congruence ( i.e. , equivalence relationcompatible with the product) on ( T n , ⋆ ) that contains all pairs of the form( t ⋆ ( t ⋆ t ) , ( t ⋆ t ) ⋆ ( t ⋆ t )) . 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 LD-system based on { x , ..., x n } . EFT-GARSIDE CATEGORIES, SELF-DISTRIBUTIVITY, AND BRAIDS 19
LD-expansions.
The relation = LD is a complicated object, about which manyquestions remain open. In order to investigate it, it proved useful to introducethe subrelation of = LD that corresponds to applying the LD-law in the expandingdirection only. Definition 4.4.
Let t, t ′ be terms. We say that t ′ is an atomic LD-expansion of t ,denoted t → LD t ′ , if t ′ is obtained from t by replacing some subterm of the form t ⋆ ( t ⋆ t ) with the corresponding term ( t ⋆ t ) ⋆ ( t ⋆ t ). We say that t ′ isan LD-expansion of t , denoted t → LD t ′ , if there exists a finite sequence of terms t , ..., t p satisfying t = t , t p = t ′ , and t i − → LD t i for 1 i p .By definition, being an LD-expansion implies being LD-equivalent, but the con-verse is not true. For instance, the term ( x ⋆ x ) ⋆ ( x ⋆ x ) is an (atomic) LD-expansionof x⋆ ( x⋆x ), but the latter is not an LD-expansion of the former. However, it shouldbe clear that = LD is generated by → LD , so that two terms t, t ′ are LD-equivalent ifand only if there exists a finite zigzag t , t , ..., t p satisfying t = t , t p = t ′ , and t i − → LD t i LD ← t i +1 for each odd i .The first nontrivial result about LD-equivalence is that the previous zigzags mayalways be assumed to have length two. Proposition 4.5. [17]
Two terms are LD-equivalent if and only if they admit acommon LD-expansion.
This result is similar to the property that, if a monoid M satisfies Ore’s condi-tions—as the braid monoid B + n does for instance—then every element in the uni-versal group of M can be expressed as a fraction of the form ab − with a, b in M .Proposition 4.5 plays a fundamental role in the sequel, and we need to recall someelements of its proof. Definition 4.6. [17] First, a binary operation ⊙ ⋆ on terms is recursively defined by(4.1) t ⊙ ⋆ x i = t ⋆ x i , t ⊙ ⋆ ( t ⋆ t ) = ( t ⊙ ⋆ t ) ⋆ ( t ⊙ ⋆ t ) . Next, for each term t , the term φ ( t ) is recursively defined by (4.2) φ ( x i ) = x i φ ( t ⋆ t ) = φ ( t ) ⊙ ⋆ φ ( t ) . 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 expressionof t in terms of variables. Examples are given in Figure 4. A straightforwardinduction shows that t ⊙ ⋆ t ′ is always an LD-expansion of t ⋆ t ′ and, therefore, that φ ( t ) is an LD-expansion of t . φ ! = t t φ ( t ) φ ( t ) φ ( t ) φ ( t ) φ ( t ) Figure 4.
The fundamental LD-expansion φ ( t ) of a term t , recursivedefinition: φ ( t ⋆ t ) is obtained by distributing φ ( t ) everywhere in φ ( 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 In [17] and [18], ∂ is used instead of φ , an inappropriate notation in the current context. φ ! = x x x x x x x x x x x x Figure 5.
The fundamental LD-expansion φ ( t ) of a term t : an example; t = x ⋆ ( x ⋆ ( x ⋆ x )) implies φ ( t ) = x ⊙ ⋆ ( x ⊙ ⋆ ( x ⊙ ⋆ x )) . braid ∆ n with respect to Artin’s generators σ i —which makes it natural to call φ ( t )the fundamental LD-expansion of t . Lemma 4.7. [17] [18, Lemmas V.3.11 and V.3.12] ( i ) The term φ ( t ) is an LD-expansion 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 4.7is established, an easy induction on d shows that, if there exists a length d sequenceof atomic LD-expansions connecting t to t ′ , then φ 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). (cid:3) The category LD + . A category (and a quiver) is naturally associated withevery graph, and the previous results invite to introduce the category associatedwith the LD-expansion relation → LD . Definition 4.8.
We denote by LD + the category whose objects are terms, andwhose morphisms are pairs of terms ( t, t ′ ) satisfying t → LD t ′ .By construction, the category LD + is left- and right-cancellative, and Proposi-ion 4.5 means that any two morphisms of LD + with the same source admit acommon right-multiple. Moreover, a natural candidate for being a left-Garsidemap is obtained by defining ∆( t ) = ( t, φ ( t )) for each term t . Question 4.9. Is LD + a left-Garside category? Question 4.9 is currently open. We shall see in Section 6.3 that it is one of themany forms of the so-called Embedding Conjecture. The missing part is that wedo not know that least common multiples exist in LD + , the problem being that wehave no method for proving that a common LD-expansion of two terms is possiblya least common LD-expansion.5. The monoid LD + and the category LD + The solution for overcoming the above difficulty consists in developing a moreprecise study of LD-expansions that takes into account the position where the LD-law is applied. This leads to introducing a certain monoid LD + whose elementsprovide natural labels for LD-expansions, and, from there, a new category LD + , ofwhich LD + is a projection. This category LD + is the one on which a left-Garsidestructure 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 bespecified by describing the path that connects the root of the tree to the root of EFT-GARSIDE CATEGORIES, SELF-DISTRIBUTIVITY, AND BRAIDS 21 t D α t • D α α αt /α t /α t /α t /α t /α t /α t /α ↑↑ Figure 6.
Action of D α 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 /α ⋆ ( t /α ⋆ t /α ) ,with ( t /α ⋆ t /α ) ⋆ ( t /α ⋆ t /α ) ; in other words, the light grey subtree isduplicated 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 sequenceof 0’s and 1’s, according to the convention that 0 means “forking to the left” and 1means “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 whoseroot has address α , if it exists, i.e. , if α is short enough.So, for instance, if t is the tree x ⋆ ( x ⋆ x ), we have t / = x , t / = x , whereas t / 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 theposition α , i.e. , replacing the subterm t /α , which is t / ⋆ ( t / ⋆ t / ), with the term( t / ⋆ t / ) ⋆ ( t / ⋆ t / ).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 LD-expansions are compositions of finitely many atomic LD-expansions, hence of D α -expansions, it is natural to use finite sequences of D α to label LD-expansions. Inother words, we extend the (partial) action of D α on terms into a (partial) actionof 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 theLD-law (in the expanding direction) at the positions α , then β , then γ . Lemma 5.2.
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.Proof. Conditions ( i ) and ( ii ) of Definition 1.7 follow from the construction. Thepoint is to prove ( iii ), i.e. , to prove that, if w , ..., w n are arbitrary finite sequencesof letters D α , then there exists at least one term t such that t • w i is defined foreach i . This is what [18, Proposition VII.1.21] states. (cid:3) The monoid LD + . There exist clear connections between the action of vari-ous D α ’s: different sequences may lead to the same transformations of trees. Ourapproach will consist in identifying a natural family of such relations and introduc-ing the monoid presented by these relations. D D ǫ D D D ǫ D ǫ D D Figure 7.
Relations between D α -expansions: the critical case. We readthat the action of D ǫ D D ǫ and D D ǫ D D coincide. Lemma 5.3.
For all α, β, γ , the following pairs have the same action on trees: ( i ) D α β D α γ and D α γ D α β ; (“parallel case”) ( ii ) D α β D α and D α D α β D α β ; (“nested case 1”) ( iii ) D α β D α and D α D α β ; (“nested case 2”) ( iv ) D α β D α and D α D α β ; (“nested case 3”) ( v ) D α D α and D α D α D α D α . (“critical case”)Sketch of proof. The commutation relation of the parallel case is clear, as the trans-formations involve disjoint subterms. The nested cases are commutation relationsas well, but, because one of the involved subtree is nested in the other, it may bemoved, and even possibly duplicated when the main expansion is performed, sothat the nested expansion(s) correspond to different names before and after themain expansion. Finally, the critical case is specific to the LD-law, and there is noway to predict it except the verification, see Figure 7. (cid:3)
Definition 5.4.
Let R LD be the family of all relations of Lemma 5.3. We defineLD + to be the monoid h{ D α | α ∈ A } | R LD i + .Lemma 5.3 immediately implies Proposition 5.5.
The partial action of the free monoid { D α | α ∈ A } ∗ on termsinduces a well defined partial action of the monoid LD + . For t a term and a in LD + , we shall naturally denote by t • a the common valueof 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 elementof LD + , thus becoming a labelled LD-expansion. However, we do not claim thata labelled LD-expansion are the same as an LD-expansion. Indeed, we do notclaim that the relations of Lemma 5.3 exhaust all possible relations between theaction 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 severallabelled expansions with different labels. As we shall see below, the uniqueness ofthe labelling is another form of the above mentioned Embedding Conjecture.5.3. The category LD + . We are now ready to introduce our main subject ofinterest, namely the category LD + of labelled LD-expansions. The starting point isthe same as for LD + , but the difference is that, now, we explicitly take into accountthe way of expanding the source is expanded into the target. EFT-GARSIDE CATEGORIES, SELF-DISTRIBUTIVITY, AND BRAIDS 23
Definition 5.7.
We denote by LD + the category whose objects are terms, andwhose 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 themorphisms of a category are triples, the source is the first entry, and the target isthe the last entry. So, for instance, a typical morphism in LD + is the triple , D ǫ D , ! , , whose source is the term x ⋆ ( x ⋆ ( x ⋆ x )) (we the default convention that unspecifiedvariables 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-Garsidecategory. To this end, we need to define the ∆-morphisms. As planned in Sec-tion 4.3, the latter will be constructed using the LD-expansions ( t, φ ( t )). Defininga labelled version of this expansion means fixing some canonical way of expandinga term t into the corresponding term φ ( t ). A natural solution then exists, namelyfollowing the recursive definition of the operations ⊙ ⋆ and φ .For w a word in the letters D α , we denote by sh ( w ) the word obtained byreplacing each letter D α of w with the corresponding letter D α , i.e. , by shifting allindices by 0. Similarly, we denote by sh γ ( w ) the word obtained by appending γ onthe left of each address in w . The LD-relations of Lemma 5.3 are invariant undershifting: if w and w ′ represent the same element a of LD + , then, for each γ , thewords 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 = ( t ⋆ t ) • sh ( a ) holdsas well, since the 0-subterm of t ⋆ t is t , whereas that of t ′ ⋆ t is t ′ . Definition 5.8.
For each term t , the elements δ t and ∆ t of LD + are defined bythe recursive rules δ t = ( t of size 1, i.e. , when t is a variable x i , D ǫ · sh ( δ t ) · sh ( δ t ) for t = t ⋆ t .(5.1) ∆ t = ( t of size 1,sh (∆ t ) · sh (∆ t ) · δ φ ( t ) for t = t ⋆ t .(5.2) Example 5.9.
Let t be x⋆ ( x⋆ ( x⋆x )). Then t / is x , and, therefore, ∆ t / is 1. Next, t / is x ⋆ ( x ⋆ x ), so (5.2) reads ∆ t = sh (∆ t / ) · δ φ ( t / ) . Then φ ( t / ) is ( x ⋆ x ) ⋆ ( x ⋆ x ).Applying (5.1), we obtain δ φ ( t / ) = D ǫ · sh ( δ x⋆x ) · sh ( δ x⋆x ) = D ǫ D D . On the other hand, using (5.2) again, we fing∆ t / = sh (∆ x ) · sh (∆ x⋆x ) · δ x⋆x = 1 · · D ǫ = D ǫ , and, finally, we obtain ∆ t = D D ǫ D D . According to the defining relations ofthe monoid LD + , this element is also D ǫ D D ǫ . Note the compatibility of the resultwith the examples of Figures 5 and 7. Lemma 5.10.
For all terms t , t , we have ( t ⋆ t ) • δ t = t ⊙ ⋆ 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 pre-cisely, we describe the simple connection existing between the category LD + andthe positive braid category B + of Example 1.9. Lemma 5.11.
Define π : { D α | α ∈ A } → { σ i | i > } ∪ { } by (5.5) π ( D α ) = ( σ i +1 if α is the address i , i.e. , ... , i times , 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 ontoa braid equivalence. All relations involving addresses that contain at least one 0collapse to mere equalities. The remaining relations are D i D j = D j D i with j > i + 2 , which projects to the valid braid relation σ i − σ j − = σ j − σ i − , and D i D j D i = D j D i D j D i , with j = i + 1 , which projects to the not less valid braid relation σ i − σ j − σ i − = σ j − σ i − σ j − . (cid:3) 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 ofthese type. For such categories, natural functors arise from morphisms between theinvolved monoids, and we fix the following notation. Definition 5.12.
Assume that
M, M ′ are monoids acting on sets X and X ′ , re-spectively. 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 ⋆ t ) = ht( t ) + 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 treeor, equivalently, the number of final )’s in t viewed as a bracketed expression. Proof.
Assume that ( t, a, t ′ ) belongs to H om ( LD + ). Put n = ht( t ). The LD-lawpreserves the right-height of terms, so we have ht( t ′ ) = n as well. The hypothesisthat t • a exists implies that the factors D 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 . (cid:3) Moreover, a simple relation connects the elements ∆ t of LD + and the braids ∆ n . EFT-GARSIDE CATEGORIES, SELF-DISTRIBUTIVITY, AND BRAIDS 25
Proposition 5.14.
We have π (∆ t ) = ∆ n whenever t has right-height n > .Proof. We first prove that ht( t ) = n implies(5.7) π ( δ t ) = σ σ ...σ n using induction on the size of t . If t is a variable, we have ht( t ) = 0 and δ t = 1, sothe equality is clear. Otherwise, write t = t ⋆ t . By definition, we have δ t = D ǫ · sh ( δ t ) · sh ( δ t ) . 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 , and π (sh ( a )) = sh( π ( a )) holds for each a in LD + . Hence, using the induction hypothesis π ( δ t ) = σ ...σ n − , we deduce π ( δ t ) = σ · · sh( σ ...σ n − ) = σ ...σ n , which is (5.7). Put ∆ = 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 = 0and ∆ t = 1, as expected. Otherwise, write t = t ⋆ t . The definition gives∆ t = sh (∆ t ) · sh (∆ t ) · δ φ ( t ) . As above, π collapses the term in the image of sh , and it transforms sh into sh.Hence, using the induction hypothesis π (∆ t ) = ∆ n − and (5.7) for φ ( t ), whoseright-height is that of t , we obtain π (∆ t ) = 1 · sh(∆ n − ) · σ σ ... σ n − = ∆ n . (cid:3) 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 pre-serving functor of LD + onto the positive braid category B + . Theorem 6.2.
Unless the category LD + is not regular, the Embedding Conjectureof [18, Chapter IX] is true. Recognizing left-preGarside monoids.
Owing to Proposition 1.11 and tothe construction of LD + from the partial action of the monoid LD + on terms, thefirst part of Theorem 6.1 is a direct consequence of Proposition 6.3.
The monoid LD + equipped with its partial action on terms viaself-distributivity is a locally left-Garside monoid with associated left-Garside se-quence (∆ 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 ),we have an easy sufficient condition when the action turns out to be monotonousin the following sense. Proposition 6.4.
Assume that M has a partial action on X and there exists amap µ : X → N such that a = 1 implies µ ( x • a ) > µ ( x ) . Then M satisfies ( LG ) .Proof. Assume that ( a , ..., a ℓ ) is a ≺ -increasing sequence in Div( a ). By definitionof a partial action, there exists x in X such that x • a is defined, and this impliesthat x • a i is defined for each i . Next, the hypothesis that ( a , ..., a ℓ ) is ≺ -increasingimplies that there exist b , ..., b ℓ = 1 satisfying a i = a i − b i for each i . We find µ ( x • a i ) = µ (( x • a i − ) • b i ) > µ ( x • a i − ) , and the sequence ( µ ( x • a ) , ..., µ ( x • a ℓ )) is increasing. As µ ( x • a ) > µ ( x ) holds,we deduce ℓ µ ( x • a ) − µ ( x ) + 1 and, therefore, M satisfies ( LG ). (cid:3) As for conditions ( LG ) and ( LG ), we appeal to the subword reversing methodof [21]. If S is any set, we denote by S ∗ the set of all finite sequences of elementsof S , i.e. , of words on the alphabet S . Then S ∗ equipped with concatenation is afree monoid. We use ǫ for the empty word. Definition 6.5.
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 b C the unique (possibly partial) map of S ∗ × S ∗ to S ∗ thatextends C and obeys the recursive rules(6.1) b C ( u, v v ) = b C ( u, v ) b C ( b C ( v , u ) , v ) , b C ( v v , u ) = b C ( v , b C ( u, v )) . Proposition 6.6 ([21] or [18, Prop. II.2.5.]) . Assume that M is a monoid satisfy-ing ( LG ) and admitting the presentation h S, R C i + , 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) b C ( b C ( b C ( a, b ) , b C ( a, c )) , b C ( b C ( b, a ) , b C ( b, c ))) = ǫ. Proof of Therem 6.1.
We shall now prove that the monoid LD + equipped itspartial action on terms via left self-distributivity satisfies the criteria of Section 6.1.Here, and in most subsequent developments, we heavily appeal to the results of [18],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 ofinner nodes in the associated binary tree. Then the hypothesis of Proposition 6.4clearly holds: if t ′ is a nontrivial LD-expansion of t , then the size of t ′ is larger thanthat of t . Then, by Proposition 6.4, LD + satisfies ( LG ).Next, we observe that the presentation of LD + in Definition 5.4 is associated witha complement on the set { D α | α ∈ A } . Indeed, for each pair of addresses α, β , thereexists in the list R LD exactly one relation of the type D α ... = D β ... . Hence, in viewof Proposition 6.6, and because we know that LD + satisfies ( LG ), it suffices to checkthat (6.2) holds in LD + for each triple D α , D β , D γ . This is Proposition VIII.1.9of [18]. Hence LD + satisfies ( LG ) and ( LG ), 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 isdefined. Then Lemma VII.3.16 of [18] states that D α is a left-divisor of ∆ t in LD + ,whereas Lemma VII.3.17 of [18] states that ∆ t is a left-divisor of D α ∆ t • D α . HenceCondition ( LG ℓoc ) of Definition 1.10 is satisfied, and the sequence (∆ t ) t ∈ T is a EFT-GARSIDE CATEGORIES, SELF-DISTRIBUTIVITY, AND BRAIDS 27 left-Garside sequence in LD + . Hence LD + is a locally left-Garside monoid, whichcompletes the proof of Proposition 6.3.By Proposition 1.11, we deduce that LD + , which is C (LD + , T ) by definition, isa 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 5.13that [ π, ht] is a surjective functor of LD + onto B + , and it just remains to provethat 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 ofaddresses 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 [18], theword b C ( u, v ) exists, and u b C ( u, v ) represents lcm( a, b ). Then π ( u b C ( u, v )) representsa common right-multiple of the braids π ( a ) and π ( b ), and, by (6.3), we have π ( u b C ( u, v )) = π ( u ) b C ( π ( u ) , π ( v )) . This shows that the braid represented by π ( u b C ( u, v )), which is π (lcm( a, b )) bydefinition, is the right-lcm of the braids π ( a ) and π ( b ). So the morphism π preservesright-lcm’s, and the proof of Theorem 6.1 is complete. (cid:3) The Embedding Conjecture.
From the viewpoint of self-distributive alge-bra, the main benefit of the current approach might be that its leads to a naturalprogram for possibly establishing the so-called Embedding Conjecture. This con-jecture, 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 + 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 actionof D α ’s by self-distributivity. Each of the above properties is conjectured to be true: this is the
EmbeddingConjecture .We turn to the proof of Theorem 6.2. So our aim is to show that the EmbeddingConjecture is true whenever the category LD + is regular. To this end, we shall usesome technical results from [18], plus the following criterion, which enables one toprove right-cancellability by only using simple morphisms. Proposition 6.8.
Assume that C is a left-Garside category and the associatedfunctor φ is injective on O bj ( C ) . Then the following are equivalent: ( i ) H om ( C ) admits right-cancellation; ( ii ) The functor φ is injective on H om ( C ) .Moreover, if C is regular, ( i ) and ( ii ) are equivalent to ( iii ) The functor φ is injective on simple morphisms of C . Proof.
Assume that f, g are morphisms of C that satisfy φ ( f ) = φ ( g ). As φ is afunctor, we first deduce φ ( ∂ f ) = ∂ ( φ ( f )) = ∂ ( φ ( g )) = φ ( ∂ g ) , hence ∂ f = ∂ g as φ is injective on objects. A similar argument gives ∂ f = ∂ g .Then, (2.1) gives f ∆( ∂ f ) = ∆( ∂ f ) φ ( f ) = ∆( ∂ g ) φ ( g ) = g ∆( ∂ g ) = g ∆( ∂ f ) . If we can cancel ∆( ∂ 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 ∆( ∂ h ) = g ∆( ∂ h ). As we have ∂ f = ∂ h = ∂ g by hypothesis, applying (2.1) gives∆( ∂ f ) φ ( f ) = f ∆( ∂ f ) = g ∆( ∂ g ) = ∆( ∂ g ) φ ( g ) = ∆( ∂ f ) φ ( g ) , hence φ ( f ) = φ ( g ) by left-cancelling ∆( ∂ f ). If ( ii ) holds, we deduce f = g , i.e. , h is right-cancellable. As simple morphisms generate H om ( C ), we deduce that everymorphism 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 , ..., f d ) and ( g , ..., g e ) be the normal forms of f and g , respectively. The regularity assumption implies that every length 2 subse-quence of ( φ ( f ) , ..., φ ( f d )) and ( φ ( g ) , ..., φ ( g e )) is normal. Moreover, ( iii ) guaran-tees that φ ( f i ) and φ ( g j ) is nontrivial. Hence ( φ ( f ) , ..., φ ( f d )) and ( φ ( g ) , ..., φ ( g e ))are normal. As φ is a functor, we have φ ( f ) ...φ ( f d ) = φ ( f ) = φ ( g ) = φ ( g ) ...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 . (cid:3) 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 resultis obvious if t has size 0. Assume t = t ⋆ t . By construction, the term φ ( t ) isobtained by substituting every variable x i occurring in the term φ ( t ) with theterm φ ( t ) ⋆ x i . Hence φ ( t ) is the 1 n − φ ( t ), where n is the commonright-height of t and φ ( t ). From there, φ ( t ) can be recovered by replacing thesubterms φ ( t ) ⋆ x i of φ ( t ) by x i . Then, by induction hypothesis, t and t , hence t ,can be recovered from φ ( t ) and φ ( t ). (cid:3) 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 [18], we de-duce 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 [18] implies a = a ′ provided a or a ′ is simple. (cid:3) We can now complete the argument.
Proof of Theorem 6.2.
The category LD + is left-Garside, with an associated func-tor φ that is injective both on objects and on simple morphisms. By Proposition 6.8, EFT-GARSIDE CATEGORIES, SELF-DISTRIBUTIVITY, AND BRAIDS 29 if LD + is regular, then H om ( LD + ) admits right-cancellation, which is one of theforms of the Embedding Conjecture, namely ( ii ) in Proposition 6.7. (cid:3) A program for proving the regularity of LD + . At this point, we are leftwith 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 possiblyproving 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 senseof 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 consequenceof Proposition VIII.5.1 of [18]. Indeed, the latter states that an element a is aleft-divisor of some element ∆ t if and only if a can be represented by a word inthe letters D α that has a certain special form. This property does not involve theterm t , and it implies that, if a left-divides ∆ t , then it automatically left-dividesevery element ∆ t ′ such that t ′ • a is defined. (cid:3) So, according to Proposition 3.10, we obtain a well defined notion of a simpleelement in LD + : an element a of LD + is called simple if it left-divides at least oneelement of the form ∆ t . Then simple elements form a seed in LD + , and are eligiblefor a normal form satisfying the general properties described in Section 3. In thiscontext, 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 ∂ f = ∂ g = t . Bydefinition, f has the form ( t, a, t • a ) for some a satisfying a ∆ t , hence simplein LD + . Similarly, f has the form ( t, b, t • b ) for some simple element b , and wehave 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 )), whencegcd( 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. (cid:3) Example 6.14.
Assume a = D ǫ , b = D , and t = x⋆ ( x⋆ ( x⋆ x )). Then t • a and t • b are defined. On the other hand, we have φ ( t ) = (( x ⋆ x ) ⋆ ( x ⋆ x )) ⋆ (( x ⋆ x ) ⋆ ( x ⋆ x )).An easy computation gives φ t ( D ǫ ) = D D and φ t ( D ) = D ǫ , see Figure 8. Wefind 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 ) , φ t ( b )) = φ t (lcm( a, b )) · D D : t t • D φ ( t ) φ ( t • D ) D D ǫ D ǫ D ǫ D D D D D D D D ǫ t • D ǫ t • lcm( D ǫ ,D ) φ ( t • D ǫ ) φ ( t ) • lcm( D D ,D ǫ ) φ ( t • lcm( D ǫ ,D )) D D ǫ D ǫ D D ǫ = Figure 8.
The left diagram shows an instance of Relation (6.4) :for the considered choice of t , we find ∆ t = D ǫ D D ǫ , ∆ t • D = D D ǫ D D D D , leading to φ t ( D ) = D ǫ and φ t ( D ǫ ) = D D . Here φ t ( D ǫ ) and φ t ( D ) are left-coprime, so (6.4) is true. The right diagramshows that the counterpart involving lcm’s fails. the terms φ ( t • D ǫ ) and φ ( t • D ) admit a common LD-expansion that is smallerthan φ t ( t • lcm( D ǫ , D )), which turns out to be φ ( t ), see Figure 8 again.The reader may similarly check that (6.4) holds for t = ( x⋆ ( x⋆x )) ⋆ ( x⋆ ( x⋆x )) with a = D and b = D ; the values are φ t ( D ) = D D D D and φ t ( D ) = D ǫ .Proposition 6.13 leads to a realistic program that would reduce the proof of theEmbedding Conjecture to a (long) sequence of verifications. Indeed, it is shown inProposition VIII.5.15 of [18] that every simple element a of LD + admits a uniqueexpression of the form a = > Y α ∈ A D ( e α ) α , where D ( e ) α denotes D α D α ...D α e − and > refers to the unique linear ordering of A satisfying α > α β > α γ for all α, β, γ . In this way, we associate with every simpleelement a of LD + a sequence of nonnegative integers ( e α ) α ∈ A that plays the role ofa 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. Remark.
Contrary to the braid relations, the LD-relations of Lemma 5.3 are notsymmetric. However, it turns out that the presentation of LD + is also associatedwith 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 coun-terpart 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) usingsome version of Proposition 6.6. EFT-GARSIDE CATEGORIES, SELF-DISTRIBUTIVITY, AND BRAIDS 31 Reproving braids properties
Proposition 5.13 and Theorem 6.1 connect the Garside structures associatedwith self-distributivity and with braids, both being previously known to exist. Inthis 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 aGarside structure, and we only know about the Garside structure of LD + .7.1. Projections.
We begin with a general criterion guaranteeing that the projec-tion 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 homomorphismof S ∗ to S ∗ that extends π , defined by π ( s ...s ℓ ) = π ( s ) ...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 ) ; • π : S → S ∪ { ǫ } satisfies π ( S ) ⊇ S and (7.1) For all a, b in S , we have C ( π ( a ) , π ( b )) = π ( C ( a, b )) .Then M is left-preGarside, and π induces a surjective right-lcm preserving ho-momorphism of M onto M .Proof. An easy induction shows that, if u, v are words on S and b C ( u, v ) exists, then b C ( π ( u ) , π ( v )) exists as well and we have(7.2) b C ( π ( u ) , π ( v )) = π ( b C ( u, v )) . 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 implicationof Proposition 6.6, the relation (6.2) involving a, b, c is true in S ∗ . Applying π andusing (7.2), we deduce that the relation (6.2) involving a, b, c is true in S ∗ . Then, as M satisfies ( LG ) by hypothesis, the converse implication of Proposition 6.6 impliesthat M is left-preGarside.Then, by definition, the relations aC ( a, b ) = bC ( b, a ) with a, b ∈ S make apresentation 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 ))in M , which shows that the homomorphism of S ∗ to M that extends π induces awell defined homomorphism of M to M . This homomorphism, still denoted π , issurjective since, by hypothesis, its image includes S .Finally, we claim that π preserves right-lcm’s. The argument is almost thesame as in the proof of Theorem 6.1, with the difference that, here, we do notassume 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 [18], the word b C ( u, v ) exists, and u b C ( u, v ) represents lcm( a, b ). Then the word π ( u b C ( u, v )) represents a commonright-multiple of π ( a ) and π ( b ) in M , and, by (7.2), we have π ( u b C ( u, v )) = π ( u ) b C ( π ( u ) , π ( v )) , which shows that the element represented by π ( u b C ( u, v )), which is π (lcm( a, b )) bydefinition, is the right-lcm of π ( a ) and π ( b ) in M . (cid:3) We turn to locally left-Garside monoids, i.e. , we add partial actions in the pic-ture. Although lengthy, the following result is easy. It just says that, if M is alocally left-Garside monoid, then its image under a projection that is compatiblewith 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 actionon X and satisfies ( LG ) ; • π : S → S ∪ { ǫ } satisfies (7.2) , θ : S → S is a section for π , ̟ : X → X is asurjection, 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 islocally left-Garside, with associated left-Garside sequence (∆ x ) x ∈ X , and π inducesa surjective right-lcm preserving homomorphism of M onto M .Proof. First, the hypotheses of Lemma 7.1 are satisfied, hence M is left-preGarsideand 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. Itremains to check that the sequence (∆ x ) x ∈ X is a left-Garside sequence with respectto 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 atomin M . So the only possiblity is a ′ = θ ( a ), i.e. , we have θ ( a ) ∆ x . Applying π , wededuce 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 anarbitrary element a for which x • a exists then follows from an easy induction onthe length of an expression of a as a product of elements of S . (cid:3) It should then be clear that, under the hypotheses of Proposition 7.2, [ π, ̟ ] is asurjective, right-lcm preserving functor of C ( M, X ) to C ( M , X ).7.2.
The case of LD + and B + . Applying the criterion of Section 7.1 to the cate-gories LD + and B + is easy. EFT-GARSIDE CATEGORIES, SELF-DISTRIBUTIVITY, AND BRAIDS 33
Proposition 7.3.
The monoid B + ∞ is a locally left-Garside monoid with respect toits 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 withthe LD-relations of Lemma 5.3, and by C the complement on { σ i | i > } associatedwith the braid relations of (1.2). We consider the maps π of Lemma 5.11, and htfrom terms to nonnegative integers. Finally, we define θ by θ ( σ i ) = D i − i . We claimthat 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. Forinstance, we find π ( C ( D , D ǫ )) = π ( D ǫ D D ) = σ σ = C ( σ , σ ) = C ( π ( D ) , 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 actionof braids on N is trivial, so (7.3) is clear. Next, define θ by θ ( σ i ) = D i − . Then θ isa section for π , and we observe that t • θ ( σ i ) is defined if and only if the right-heightof 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 itdepends on ht( t ) only. So (7.5) is satisfied.Therefore, Proposition 7.2 applies, and it gives the expected result. (cid:3) 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 locallyleft-Garside. Point ( ii ) follows from Proposition 1.12 since, for each n , the sub-monoid B + n is the monoid ( B + ∞ ) n in the sense of Definition 1.7. (cid:3) Thus, as announced, the Garside structure of braids can be recovered from theleft-Garside structure associated with the LD-law.8.
Intermediate categories
We conclude with a different topic. The projection of the self-distributivity cat-egory LD + to the braid category B + described above is partly trivial in that termsare involved through their right-height only and the corresponding action of braidson integers is just constant. Actually, one can consider alternative projectionscorresponding 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 ofintegers via their permutations. Indeed, the rule(8.1) ( x , ..., x n ) • σ i = ( x , ..., x i − , 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 cate-gory C ( B + ∞ , N ∗ ), which clearly projects to B + .We shall now describe an explicit projection of LD + onto this category. Werecall that terms have been defined to be bracketed expressions constructed froma fixed sequence of variables x , x , ... (or as binary trees with leaves labelled with b π b πx p x q x p x q ( ..., p, q, ... ) ( ..., q, p, ... ) σ i D i − Figure 9.
Compatibility of the action of D i − on sequences of “sub-right” variables and of the action of σ i on sequences of integers. variables x p ), and that, for t a term and α a binary address, t /α denotes the subtermof t whose root, when t is viewed as a binary tree, has address α . Proposition 8.1.
Let b B + be the category associated with the partial action (8.1) of B + ∞ on N ∗ . Then b B + is a Garside category, and the projection [ π, ht] of LD + onto B + factors through b B + into LD + [ π, b π ] −−−−→ b B + [id , lg] −−−−−→ B + , where b π is defined for ht( t ) = n by b π ( t ) = (var R ( t / ) , var R ( t / ) , ..., var R ( t / n − )) , var R ( t ) denoting the index of the righmost variable occurring in t . So, a typical morphism of b B + is ((1 , , , σ , (2 , , x p x p x p n ( p , p , ..., p n ). b π Proof (Sketch).
The point is to check that the action of the LD-law on the indices ofthe right variables of the subterms with addresses 1 i D i − ,and the expected relation is shown in Figure 9. Details are easy. Note that, forsymmetry reasons, the category b B + is not only left-Garside, but even Garside. (cid:3) Action of braids on LD-systems.
The action of positive braids on se-quences of integers defined in (8.1) is just one example of a much more generalsituation, namely the action of positive braids on sequences of elements of any LD-system. It is well known—see, for instance, [18, Chapter I]—that, if (
S, ⋆ ) is anLD-system, i.e. , ⋆ is a binary operation on S that obeys the LD-law, then(8.2) ( x , ..., x n ) • σ i = ( x , ..., x i − , 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, apartial action of B + ∞ on the set S ∗ of all finite sequences of elements of S . EFT-GARSIDE CATEGORIES, SELF-DISTRIBUTIVITY, AND BRAIDS 35
Proposition 8.2.
Assume that ( S, ⋆ ) is an LD-system, and let B + S be the categoryassociated with the partial action (8.2) of B + ∞ on S ∗ . Then B + S is a left-Garsidecategory, and, for all s , s , ... in S , the projection [ π, ht] of LD + onto B + factorsthrough 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 / ) , ..., eval S ( t / n − )) , 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 thecategory b B + of Proposition 8.1. In this case, the (partial) action of braids is notconstant as in the case of B + , but it factors through an action of the associatedpermutations, 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 sh( x ) − , where we recallsh 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 ′ holdsfor at least one sequence s in ( B ∞ ) ∗ : this follows from Lemma III.1.10 of [23]. Thissuggests 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 algebraiclaws as well. However, at least from the viewpoint of Garside structures, the caseof self-distributivity seems quite particular.
The case of associativity.
Associativity is the law x ( yz ) = ( xy ) z . It is syntac-tically close to self-distributivity, the only difference being that the variable x isnot duplicated in the right hand side. Let us say that a term t ′ is an A -expansionof another term t if t ′ can be obtained from t by applying the associativity law inthe left-to-right direction only, i.e. , by iteratively replacing subterms of the form t ⋆ ( t ⋆ t ) by the corresponding term ( t ⋆ t ) ⋆ t . Then the counterpart of Propo-sition 4.5 is true, i.e. , two terms t, t ′ are equivalent up to associativity if and only ifthey admit a common A -expansio, a trivial result since every size n term t admitsas 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 + whose oblectsare terms, and whose morphisms are pairs ( t, t ′ ) with t ′ an A -expansion of t . Asin Section 5.1, we can take positions into account, using A α when associativity isapplied at address α , and introduce a monoid A + that describes the connectionsbetween the generators A α [22]. Here the relations of Lemma 5.3 are to be replacedby analogous new relations, among which the MacLane–Stasheff Pentagon relations A α = A α A α A α . The monoid A + turns out to be a well known object: indeed, itis (isomorphic to) the submonoid F + of R. Thompson’s group F generated by thestandard generators x , x , ... [11].Finally, as in Section 5, we can introduce the category A + , whose objects areterms, 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 + and A + are isomorphic; A + is left-Garside withGarside 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 Garsidestructure(s) is trivial: the maps φ and ψ are constant on each orbit of the actionof A + on terms, and it easily follows that every morphism in A + and A + is left-simpleand right-simple so that, for instance, the greedy normal form of any morphismalways has length one . The only observation worth noting is that A + provides anexample 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.
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Laboratoire de Math´ematiques Nicolas Oresme, Universit´e de Caen, 14032 Caen,France
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