Coherent presentations of structure monoids and the Higman-Thompson groups

COHERENT PRESENT A TIONS OF STR UCTURE MONOIDS AND THE HIGMAN-THOMPSON GR OUPS JONA THAN A. COHEN Abstract. Struct ure monoids and groups ar e algebraic inv ariants of equa- tional v arieties. W e s ho w how to construct presen tations of these ob jects from coheren t cat egorifications of equat ional v arieties, generalising sev eral results of Dehorno y . W e subsequen tly r ealise the higher Thompson groups F n, 1 and the H igman-Thompson groups G n, 1 as str ucture groups. W e go on to obtain presen tations of these groups via coheren t categorifications of the v ari eties of higher-order asso ciativity and of higher-order asso ciativity and commutat iv- ity , resp ectiv ely . These categorifications generalise M ac Lane’s p ent agon and hexagon conditions for coherently asso ciativ e and commutat ive bifunctors. 1. Introduction Thompson’s gr oup F is a finitely presented infinite s imple gro up that a ppea rs in a num be r o f guises. F or us, the most useful description is that of Br own [Bro 87], which casts the ele men ts o f F as pairs o f finite binary trees having the same num b er of leav es, sub ject to a cer tain equiv a lence relation on the pairs . This description suggests t hat F may in fact have something to do with asso ciativity , w ith the elements represe nting pairs of equiv alent terms in some fr ee semig roup. This ob- serv ation tur ns out to b e fruitful and Deho rnoy [Deh05 ] has explo ited it in order to rea lise F as a n algebraic inv ar iant o f the v ariety o f s e migroups and subsequently to co ns truct a “geometric” presen tation o f F . In a similar manner, Deho r noy re- alises Thompson’s gro up V as an algebra ic inv ar iant of the v ariety of commutativ e semigroups and constructs a geo metric presentation of V . The r elations in Dehor noy’s presentations consis t of tw o pa rts. First, ther e are the s o -called geometric r elations, which ar ise purely from the fact that a semigroup is, in the fir st instance, a ma gma. The second class of relations arise from the particular equational s tructure of the v ariet y at ha nd. In the case of F , one a ddi- tional class of r elations are a dded co rresp onding to the Stasheff-Mac Lane p entagon [ML63] and in the case of V , the pres entation further contains a clas s o f rela tions corres p o nding to the Mac Lane hexagon, w hich enco des the essen tial in teraction betw een as s o ciativity and commutativit y . The first goal of this pap er is to place Dehornoy’s co nstructions in a mo r e g en- eral co nt ext. More pr e cisely , instead of a set with op era tions and eq uations, we consider a categ ory with functor s and na tur al isomor phisms. Within this setting, Dehornoy’s geometric relations cor resp ond to functor iality and naturality of the a s- so ciated catego rical structure. The second cla s s of re lations corres p o nd to so-called “coherence axio ms”, which a re a collection of equations making the fr ee categ orical structure equiv alent to a preor der. Dehornoy’s relation of F and V to particular equational v arieties is a sp ecia l case of a more gener a l co nstruction [Deh93] assoc iating an inv erse mono id to a ny balanced equational v ariety . This mo noid is termed the “ structure monoid” o f the v ariety . In Section 2, we begin by r ecalling the construction from [Deh93]. W e Date : Nov em b er 7, 2018. 1 2 JONA THAN A. COHEN then go on to desc r ib e “ca tegorifica tions” of equational v arieties a nd show that a coherent categor ific a tion of an equa tional v ariety g ives rise to a pr esentation of the asso ciated structure monoid. In c e rtain favourable situations, the structure monoid turns out to b e a group and we show that the co nstruction of a presentation fr om a coherent catego rification car ries over to this setting. Higman [Hig7 4] has s hown that Thompson’s gro up V is in fac t part o f an infinite family of finitely presented g roups G n,r , whic h ar e e ither simple or ha ve a simple subgroup o f index 2 . Brown [Bro87] s ubsequently show ed that Thompso n’s g roup F fits into a similar infinite family F n,r . W e recall the de finitio ns of F n, 1 and G n, 1 in Section 3 . In Section 4 , we s how that the groups F n, 1 arise as the str ucture groups of n -catalan a lgebras , which enco de a notion of ass o ciativity for an n -ary function symbo l. Similarly , w e show that the groups G n, 1 arise as the structure groups of s ymmetric n -catala n algebra s, which contain a n actio n o f the s y mmetric group S n on the v ariables of an n -a ry function s ymbol. In Section 5, we construct a coherent categorifica tion of n -ca talan alg ebras, whic h we call n -catala n categor ies. The co herence ax ioms for n -catalan c a tegories dire c tly generalise the Stasheff-Mac Lane p entagon axio m, with a new class of axio ms ap- pea ring when n ≥ 3. F ollowing from the r esults of sections 2 and 4, we obtain new pres ent a tio ns for F n, 1 . In Section 6, we co nstruct s y mmetric n - c atalan cate- gories a nd s how tha t these form a c oherent catego rification of symmetr ic n -cata lan algebras , th us o btaining new pr esentations for G n, 1 . As in the cas e of n -catala n categorie s, additional cla sses o f coher ence axioms are requir ed when n ≥ 3 . Throughout this pap er, we read f · g as “ f follow ed by g ”. 2. Free ca tegories a nd structure mono ids W e beg in this section by reca lling Dehor noy’s construction of an inv erse monoid asso ciated to a balanced equatio na l theo r y [Deh93]. F ollowing this, we des crib e a pro cess for o btaining a categorical version of an equational theory and a metho d of constructing a monoid from such a ca teg orificatio n. Finally , we link the tw o constructions together by showing that coherent c a tegorifica tions give rise to pre- sentations of structure monoids. W e ba se our analysis at the level o f theories, r ather than of equational v arieties . While this is seemingly a t o dds with Dehornoy’s re- sult [Deh93] that s tructure monoids are indep endant of the par ticular equational presentation o f a v ariet y , differ ing prese nt atio ns of the same v ariety lead to distinct categorifa ctions and thence to distinct pr esentations of the structur e mo noid. 2.1. Structure monoi ds ass o ciated to equational theories. F or a g raded set of function symbols F and a s et X , we denote b y F F ( X ) the abso lutely free term algebra generated by F on X . An equationa l theory is a tuple ( V , F , E ), where V is a set o f v aria bles, F is a graded set o f function symbols and E is an equational theory on F F ( V ). A map ϕ : V → F F ( V ) is called a substitution and it extends inductively to an endomor phis m F F ( V ) → F F ( V ). By abuse of notation, we lab el this latter map by ϕ as w ell. W e us e [ V , F F ( V )] to denote the set of all substitutions. F or a term s ∈ F F ( V ) a nd a substitution ϕ ∈ [ V , F F ( V )], we use s ϕ to deno te the image o f s under ϕ . The supp ort o f a term s is the set of v ariables app ea r ing in it. A pair of terms ( s, t ) is b alanc e d if they have the same supp ort. Definition 2. 1. Given a b alanc e d p air of terms ( s, t ) in F F ( V ) , we u se ρ s,t to denote t he p artial fun ction F F ( V ) → F F ( V ) with gr aph { ( s ϕ , t ϕ ) | ϕ ∈ [ V , F F ( V )] } . F or a balanced pair of terms ( s, t ), the partial function ρ s,t is functional since the supp ort o f t is a subset of the suppor t of s . The stronger res triction that the COHERENT PRE SENT A TIONS OF STRUCTURE MONOIDS AND THE HIGMAN-THOMPSON GROUPS 3 pair is balanced is requir ed since we wish to utilise the inv erse partial function ρ t,s as well. Given an e quational theory T := ( V , F , E ), we us e [ E ] to denote the co ngruence generated by E on F F ( V ) and w e use F T ( V ) to denote the quotient F F ( V ) / [ E ]. Similarly , we use [ s ] to deno te the co ngruence clas s of a term s in F T ( V ). It is clea r that [ u ] = [ ρ s,t ( u )] for any ba la nced pa ir o f terms ( s, t ) a nd any term u ∈ dom( ρ s,t ). How ever, the co lle c tion o f all pa r tial maps ρ s,t for ( s, t ) ∈ E is not sufficien t to generate [ E ], since equations apply to subterms as well. T o this end, we intro duce translated versions of the maps ρ s,t , that apply to a rbitrary s ubterms. A subterm s o f a term t is natura lly sp ecified by the no de wher e its ro ot lies in the term tree of t , which in turn is completely s pe cified by the unique path from the ro ot o f t to the ro ot o f s in the ter m tre e . A path in a term tree may be specified by a n alterna ting se q uence of function symbo ls and num b ers, where the n umbers indicate a n argumen t of a function symbol. More forma lly , we hav e the fo llowing situation. F or a gra ded set F := ` n F n , we set A F := [ n [ F ∈F n { ( F, 1) , . . . , ( F , n ) } . The set o f addr esses asso ciate d to F is denoted by A ∗ F and is the fre e monoid generated by A F under conc a tenation, with the unit b eing the empty string λ . F or a term t ∈ F F ( V ) a nd an address α ∈ A ∗ F , we use sub( t, α ) to deno te the subterm of t at the addres s α . Note that sub( t, α ) only ex ists if the term tree of t contains the path α and that s ub( t, λ ) = t . Example 2. 2. Su pp ose t hat F := { F, G } , wher e F is a binary function symb ol and G is a ternary function symb ol. Supp ose that V is a set of variables. Then, the term t := F ( w, G ( x, y , z )) is in F F ( V ) . The term t r e e of t is: F w G x y z      ? ? ? ? ?       : : : : : : The term t has t he fol lowing su bterms: sub( t, ( F , 1 )) = w sub( t, ( F , 2 )) = G ( x, y , z ) sub( t, ( F , 1 )( G, 1) = x sub( t, ( F , 1 )( G, 2) = y sub( t, ( F , 1 )( G, 3)) = z Definition 2.3 (Orthogonal) . Given a gr ade d set F and addr esses α, β ∈ A ∗ F , we say that α and β ar e ortho gonal and write α ⊥ β if neither α nor β is a pr efix of the other. Given a term t , and addr esses α and β , the subterms sub( t, α ) and sub( t, β ) ar e ortho gonal if α ⊥ β . Our curr ent addr essing sys tem is sufficient to describe translated copies of the basic op erator s. Definition 2. 4. Given a gr ade d set of function symb ols F , a variable set V , a b alanc e d p air of terms ( s, t ) ∈ F F ( V ) and an addr ess α ∈ A ∗ F , the α -t r anslate d c opy of ρ s,t is denote d ρ α s,t and is t he p artial map F F ( V ) → F F ( V ) define d as fol lows: • A t erm u ∈ F F ( V ) is in t he domain of ρ α s,t if sub( u, α ) is define d and is in the domain of ρ s,t . 4 JONA THAN A. COHEN • F or u ∈ dom( ρ α s,t ) , t he image ρ α s,t ( u ) is define d by sub( ρ α s,t ( u ) , α ) = ρ s,t (sub( u, α )) and s ub( ρ α s,t ( u ) , β ) = sub( u, β ) for every addr ess β ortho gonal to α . Note that ρ λ s,t = ρ s,t . W e are fina lly in a p osition to introduce the s tructure monoid generated by an equational theory . Definition 2.5 (Structure Monoid) . Given an e quational the ory T := ( V , F , E ) , t he structure mo noid of T , denote d Struct( T ) , is the m onoid of p artial en domorphisms of F F ( V ) gener ate d by the fol lowing maps u nder c omp osition:  ρ α s,t | ( s, t ) or ( t, s ) ∈ E and α ∈ A ∗ F  The structure monoid of a n equationa l theory is readily seen to completely ca p- ture the equational theor y . Lemma 2.6 (Dehor noy [Deh9 3]) . L et T := ( V , F , E ) b e a b alanc e d e quational the ory and let t, t ′ ∈ F F ( V ) . Then t = T t ′ if and only if ther e is some ρ ∈ Struct( T ) such that ρ ( t ) = t ′ .  Given an equational theory T = ( V , F , E ) and maps ρ s 1 ,t 1 , ρ s 2 ,t 2 ∈ Struct( T ), the comp ositio n ρ s 1 ,t 1 · ρ s 2 ,t 2 may b e empt y . It is nonempty pr e c isely when there exist subs titutions ϕ, ψ ∈ [ V , F F ( V )] such that t ϕ 1 = s ψ 2 . In this cas e , we s ay that the pair ( t 1 , s 2 ) is unifiable and that ( ϕ, ψ ) is a unifier o f the pair. In the ca se where ( t 1 , s 2 ) is not unifiable, the compo sition ρ s 1 ,t 1 · ρs 2 , t 2 results in the empt y op erator , which we denote by ε . Note that, for any oper ator ρ ∈ Struct( T ), w e hav e ρ · ε = ε · ρ = ε . T he exis tence of the empt y op era tor makes fre ely computing with inv erses in Struct( T ) imp os s ible. Definition 2. 7 (Comp o sable) . A n e quational the ory ( V , F , E ) is comp os able if any p air of terms in S ( s,t ) ∈E { s, t } ar e unifiable. Struct( T ) a lwa ys forms an inv erse monoid [Deh06] and co ntains the empty op er- ator pr e c isely whe n T is not comp o sable. O ne wa y in which to tr a nsform Struct( G ) int o a gro up is by pa ssing to the universal gr oup of Struct( T ), which we denote by Struct G ( T ), by colla ps ing all idemp otents to 1. In the case where T is comp osa ble, the idemp otent element s of Str uct( T ) are precisely those o per ators that act a s the ident ity on their domain. A par ticular class of comp osable theor ies is pr ovided by a certa in class o f linear theories. Recall that an equation s = t is line ar if it is balanced and each v ariable app ears precisely o nce in b o th s and t . An equational theory is linear if each o f its defining equations is linear. Lemma 2 .8 (Dehorno y [Deh0 6 ]) . A line ar e quational the ory c ontaining pr e cisely one function symb ol is c omp osable.  It fo llows from the ab ov e lemma that each linea r eq uational theory containing precisely one function symbo l g ives rise to a structure gr oup . Example 2.9. The e quational the ories for semigr oups, S , and for c ommut ative semigr oups, C , ar e b oth line ar. Sinc e these the ories involve a single binary op er ator, L emma 2.8 implies that they ar e c omp osable. In t his c ase we have that Struct G ( S ) is Thomopso n ’s gr oup F and Struct G ( C ) is Thompson ’s gr oup V [Deh05] . COHERENT PRE SENT A TIONS OF STRUCTURE MONOIDS AND THE HIGMAN-THOMPSON GROUPS 5 2.2. Categorification of equational theories. An equatio nal theo ry T = ( V , F , E ) defines an algebraic s tructure on a set. In pass ing to the structure mo noid Struct( T ) we a bstract aw ay fro m the underlying s et and fo cus instea d on the partial op era tions generated by the congr uence [ E ]. This suggests passing to a structure where the op erations genera ted by E are g iven fir st-class status. In o rder to achiev e this g o al, we pass from a set with algebr a ic structur e to a ca teg ory with algebra ic structure. Definition 2. 10 (Precateg orification) . Given a b alanc e d e quational the ory T = ( V , F , E ) , t he precatego r ification of T is the stru ct ur e b T := ( b V , b F , b E ) , which c onsists of: (1) The discr ete c ate gory b V gener ate d by V . (2) F or every function symb ol F ∈ F n , a funct or b F : b V n → b V in b F . The functor b t for a term t ∈ F F ( V ) is define d inductively. (3) F or every e quation ( s, t ) ∈ E , a natur al isomorphi sm b ρ s,t : b s → b t . We use the notation b ρ t,s := ( b ρ s,t ) − 1 . A pr ecategor ification o f an equatio nal theor y should b e thought of as b eing akin to a graded set of function symbols, rather than to an equational theory . The rea son for this is that, althoug h it contains a ll of the informatio n of an equational theory , it does not cont ain enough information to ensure that it faithfully represe nts the equational structure. A preca tegorifica tion generates a ca teg ory whose ob jects are the absolutely free term algebr a a nd who s e morphisms a re “itera ted substitutions” of the ba sic maps. Before making this statement precise, we int ro duce some no ta- tion. Given a term b s ∈ F b F ( b V ), its supp ort, supp( b s ), is the set of ob jects a ppe aring in it. F or a morphism b ρ : b s → b t , if supp( b s ) = s upp( b t ) = { x 1 , . . . , x n } , then w e often wr ite b ρ ( x 1 , . . . , x n ) : b s ( x 1 , . . . , x n ) → b t ( x 1 , . . . , x n ) to sp ecific a lly refer to the ob jects in the supp o rt. Note that a particular x i may app ear more than once in b s or b t . Definition 2.1 1 . Given a pr e c ate gorific atio n b T := ( b V , b F , b E ) of a b alanc e d e qua- tional the ory ( V , F , E ) , we denote by F b T ( b V ) the c ate gory whose obje cts ar e F b F (Ob( b V )) and whose morp hisms ar e c onstru cte d inductively as fol lows: (1) F b T ( b V ) c ontains the identity 1 : b V → b V . (2) F b T ( b V ) c ontains b E . (3) If  b ρ i : b s i → b t i  n i =1 ⊂ F b T ( b V ) and b F ∈ b F n , then b F ( b ρ 1 , . . . , c ρ n ) : b F ( b s 1 , . . . , c s n ) → b F ( b t 1 , . . . , b t n ) is in F b T ( b V ) . (4) If b ρ ( x 1 , . . . , x n ) : b s ( x 1 , . . . , x n ) → b t ( x 1 , . . . , x n ) is in F b T ( b V ) and b ϕ ∈ [ b V , F b F (Ob( b V ))] is a subst itution, t hen b ρ ( x b ϕ 1 , . . . , x b ϕ n ) : b s ( x b ϕ 1 , . . . , x b ϕ n ) → b t ( x b ϕ 1 , . . . , x b ϕ n ) is in F b T ( b V ) (5) If b ρ 1 : b s → b u and b ρ 2 : b u → b t ar e in F b T ( b V ) , then b ρ 1 · b ρ 2 : b s → b t is in F b T ( b V ) . It is straig ht for ward to show that for b T := ( b V , b F , b E ), the catego r y F b T ( b V ) is the free categ ory on b V co ntaining a ll of the functor s in b F and all of the natura l transformatio ns in b E , so that F b T ( b V ) forms our analog ue of the absolutely free term algebra. Categoric a l str uctures very r arely a r ise as precatego rifications of equationa l the- ories. F ar more common is to req uire in the definition of a structure that cer tain 6 JONA THAN A. COHEN diagrams co mm ute. In particular, given a collectio n of diagrams D , each of which consists of a parallel pair of morphisms b ρ 1 , b ρ 2 : b s → b t in F b T ( b V ), w e may build a congruence [ D ] o n the set of mo rphisms of F b T ( b V ). F actoring out by this congr uence yields the free b T -structure on b V satisfying the prop er ty that all of the diag rams in D comm ute. F or particula r categoric a l structures , it is p os sible to o btain a set o f such diag r ams, who se commutativit y implies the commutativit y of any diagra m in F ( b T , D ) ( b V ) := F b T ( b V ) / [ D ]. This phenomenon is termed “coher ence” a nd it is eq uiv - alent to requiring that F ( b T , D ) ( b V ) is a preor der. Coherence w as fir st inv estigated by Mac Lane [ML63] in relation to monoidal a nd symmetric monoidal categories and has subsequently for med a ma jor pa r t of catego rical universal alge br a, with an abstract categoric al trea tment having b een provided by Kelly [Kel7 2]. Definition 2 . 12 (Categorification) . A categor ification of a b alanc e d e quational the ory T := ( V , F , E ) is a p air ( b T , D ) , wher e b T is a pr e c ate gorific ation of T and D is a c ol le ction of p ar al lel p airs of morphisms in F b T ( b V ) . We say that ( b T , D ) is coherent if F ( b T , D ) ( b V ) is a pr e or der. Example 2.13. The pr e c ate gori fic ation of the the ory of semigr oups c onsists of a binary functor ⊗ , to gether with a natur al isomorphism: α ( x, y , z ) : x ⊗ ( y ⊗ z ) → ( x ⊗ y ) ⊗ z . Mac L ane [ML63] showe d that, in or der to obtain a c oher ent c ate gorific ation of the the ory of semigr oups, we ne e d only t he“p entagon axiom”, which st ates that t he fol lowing diagr am c ommutes: a ⊗ ( b ⊗ ( c ⊗ d ))) ( a ⊗ b ) ⊗ ( c ⊗ d ) a ⊗ (( b ⊗ c ) ⊗ d ) (( a ⊗ b ) ⊗ c ) ⊗ d ( a ⊗ ( b ⊗ c )) ⊗ d α   1 ⊗ α   α   α   α ⊗ 1 o o The pr e c ate gorific ation of the t he ory of c ommut ative semigr oups has an additional natur al isomorphism τ with t he fol lowing c omp onents: τ ( x, y ) : x ⊗ y → y ⊗ x. Mac L ane [ML6 3] went on to show t hat a c oh er ent c ate gorific ation of the the ory of c ommut ative semigr oups is obtaine d via the p entagon axiom, to gether with the axiom that τ · τ = 1 and the “hexagon ax iom”, which states that the fol lowing diagr am c ommutes: a ⊗ ( b ⊗ c ) ( b ⊗ c ) ⊗ a b ⊗ ( c ⊗ a ) b ⊗ ( a ⊗ c ) ( a ⊗ b ) ⊗ c ( b ⊗ a ) ⊗ c τ / / α − 1 / / 1 ⊗ τ   α   τ ⊗ 1 / / α − 1 / / 2.3. Monoid presentations from coheren t categorifications. Dehornoy’s util- isation of the p entagon and hexagon co herence ax ioms in order to obtain pr esen- tations of Thompson’s groups [Deh05] is indica tive of a mor e genera l relations hip betw een structure monoids and coherent ca tegorifica tio ns of equational theories. COHERENT PRE SENT A TIONS OF STRUCTURE MONOIDS AND THE HIGMAN-THOMPSON GROUPS 7 The fir st s tep on the r oad to formalising this rela tionship is to constr uct a monoid presentation out o f a catego rification of an equationa l theory . The initial difficulty is to construct a set of genera tors for the monoid corr esp onding to the gener ators of the structure mono id. Definition 2.14 (Singular morphisms) . Given a b alanc e d e quational the ory T = ( V , F , E ) and a pr e c ate gorific ation b T = ( b V , b F , b E ) , the set of singular mo rphisms of F b T ( b V ) is denote d Sing( b T ) and is gener ate d as fol lows: (1) Every n atur al isomorphism in b T is singular. (2) If b ρ is singular and b ϕ is a substitu tion, then ˆ ρ b ϕ is singular. (3) If b ρ is singular, b F ∈ c F n and 1 ≤ i ≤ n , then b F ( i − 1 z }| { 1 , . . . , 1 , b ρ, n − i z }| { 1 , . . . , 1) , is singular. In essence, the set o f singular mor phisms are those that con tain pr ecisely one instance of a generating natura l iso morphism. Their suitability to a ct as generator s is hig hlighted by the following lemma. Lemma 2.15. L et b T := ( b V , b F , b E ) b e a pr e c ate gorific ation of a b alanc e d e quational the ory ( V , F , E ) . Every morphism in F b T ( b V ) is a c omp osite of finitely many singular morphisms. Pr o of. The only p otential pro ble m is a morphis m of the for m b F ( b ρ 1 , . . . , c ρ n ). How- ever, by functoriality of b F , we have b F ( b ρ 1 , . . . , c ρ n ) = b F ( b ρ 1 , 1 , . . . , 1 ) · b F (1 , b ρ 2 , 1 , . . . , 1 ) · . . . · b F (1 , . . . , 1 , c ρ n ) .  In o rder to make the r e lationship b etw een the mo no id we cons truct from a ca te- gorificatio n and the structur e mono id more per spicuous, we introduce an address ing system for singular mor phisms. Definition 2.16 (Type/Address ) . The type , T ( b ρ ) of a s ingu lar morphism b ρ of a pr e c ate gorific ation ( b V , b F , b E ) is the gener ating natu r al isomorphism that app e ars as in instanc e in it. The addr ess, A ( b ρ ) is the wor d of A ∗ b F c onstructe d as fol lows: A ( b ρ ) =      ( b F , i ) A ( b σ ) if b ρ = b F ( i − 1 z }| { 1 , . . . , 1 , b σ , 1 , . . . , 1) λ otherwise Given a categorifaca tion ( b T , D ), we can now construct a monoid who se ge n- erators a re the singular morphisms o f b T a nd whose relations are g enerated by functoriality , natura lity a nd the dia grams in D . Definition 2.17. L et T := ( V , F , E ) b e a b alanc e d e quational the ory and let ( b T , D ) b e a c ate gorific ation of T . The monoid P ( b T , D ) is the monoid gener ate d by { T ( b ρ ) A ( b ρ ) | b ρ ∈ Sing ( T ) } ∪ { ˆ ρ b α s,s | ( s, t ) or ( t, s ) in b E and b α ∈ A ∗ b F } if T is c omp osable and by { T ( b ρ ) A ( b ρ ) | b ρ ∈ Sing ( T ) } ∪ { ˆ ρ b α s,s | ( s, t ) or ( t, s ) in b E and b α ∈ A ∗ b F } ∪ { b ε } otherwise, s u bje ct t o the fol lowing r elations. 8 JONA THAN A. COHEN • Inverse: ˆ ρ b α s,t · ˆ ρ b α t,s = ˆ ρ b α s,s ˆ ρ b α s,s · ˆ ρ b α s,t = ˆ ρ b α s,t ˆ ρ b α s,t · ˆ ρ b α t,t = ˆ ρ b α s,t • Comp osi tion: If t 1 and s 2 ar e not unifiable then ˆ ρ b α s 1 ,t 1 · ˆ ρ b α s 2 ,t 2 = b ε • Empty op er ator : ˆ ρ b α s,t · b ε = b ε b ε · ˆ ρ b α s,t = b ε • F unctoriality : F or b α ⊥ b β : ˆ ρ b α s,t · ˆ ρ b β u,v = ˆ ρ b β u,v · ˆ ρ b α s,t • Natur ality: Supp ose that ˆ ρ s,t is a gener ator and t hat some variable x app e ars at addr esses c β 1 , . . . , c β p in ˆ s and at addr esses b γ 1 , . . . , b γ q in ˆ t . Then, for al l addr esses b α, b δ and e ach gener ator ˆ ρ u,v : ˆ ρ b α s,t · ˆ ρ b α c γ 1 b δ u,v · . . . · ˆ ρ b α c γ q b δ u,v = ˆ ρ b α c β 1 b δ u,v · . . . · ˆ ρ b α c β p b δ u,v · ˆ ρ b α s,t • Coher enc e: F or ( σ 1 · . . . · σ p , τ 1 , . . . , τ q ) ∈ D , wher e e ach σ i and τ j is singular, set : T ( σ 1 ) A ( σ 1 ) · . . . · T ( σ p ) A ( σ p ) = T ( τ 1 ) A ( τ 1 ) · . . . · T ( τ q ) A ( τ q ) The relations for functoriality and natur ality in P ( b T , D ) are adapted fro m [Deh06]. The functor iality r e lation is precis ely the re quirement that each op erato r b F ∈ b F is a functor. The natur ality condition is, in turn, precisely the r equirement that each b ρ ∈ b E is a natural transformation. The rather in volved addressing system in the naturality condition is due to the fac t that the same v ariable may app ear multiple times in different p ositions on either side of an equatio n. F or naturality , one needs to a pply a map to each of these instances o f the v ariable simultaneously . W e now set ab out relating P ( b T , D ) to Struct( T ). Lemma 2.18. L et T b e a b alanc e d e quational the ory and let ( b T , D ) b e a c ate gori- fic ation of T . Then P ( b T , D ) is an inverse m onoid. Pr o of. F or nonempt y ˆ ρ := ˆ ρ c α 1 s 1 ,t 1 · . . . ˆ ρ c α k s k ,t k , set ˆ ρ − 1 := ˆ ρ c α k t k ,s k · . . . ˆ ρ c α k t 1 ,s 1 . Then it follows from the Inv erse relatio ns that ˆ ρ · ˆ ρ − 1 · ˆ ρ = ˆ ρ ˆ ρ − 1 · ˆ ρ · ˆ ρ − 1 = ˆ ρ − 1 Since we also have that b ε · b ε · b ε = b ε , it follows that P ( b T , D ) forms a n inv erse monoid.  Theorem 2. 19. L et T b e a b alanc e d e quational the ory and let ( b T , D ) b e a c ate- gorific ation of T . The fol lowing map is an epimorphism of inverse monoids and it is an isomorphi sm if and only if ( b T , D ) is c oher ent: P ( b T , D ) Θ − → Struct( T ) ˆ ρ c α 1 s 1 ,t 1 · . . . · ˆ ρ c α k s k ,t k 7− → ρ α 1 s 1 ,t 1 · . . . · ρ α k s k ,t k COHERENT PRE SENT A TIONS OF STRUCTURE MONOIDS AND THE HIGMAN-THOMPSON GROUPS 9 Pr o of. By construction, Θ is a homo morphism of inv erse mo noids. F or surjectivity , we need only show tha t every g enerator ρ α s,t ∈ Str uct( T ) cor resp onds to some singu- lar mor phism S ( ρ α s,t ) ∈ b T . This sing ular mor phism can b e constr ucted recursively as follows: S ( ρ α s,t ) =      b F ( i − 1 z }| { 1 , . . . , 1 , S ( ρ β s,t ) , 1 , . . . , 1) if α = ( b F , i ) β ρ s,t if α = λ It remains to show that Θ is faithful if and only if ( b T , D ) is coherent. Suppo se that Θ is faithful and let b ρ 1 , b ρ 2 be a parallel pair of mor phis ms in b T . Then Θ( b ρ 1 ) = Θ( b ρ 2 ), since b ρ 1 and b ρ 2 hav e the same so urce and tar get. Since Θ is faithful, it follows that b ρ 1 = b ρ 2 . Conv ersely , supp ose that ( b T , D ) is coher ent a nd that Θ ( b ρ 1 ) = Θ ( b ρ 2 ). Then, b ρ 1 and b ρ 2 hav e the same source a nd targ et. Since ( b T , D ) is coher ent, it follo ws that b ρ 1 = b ρ 2 .  As in the case of s tructure monoids, when the theory T is bala nced and co m- po sable, we may construct a gr oup P G ( b T , D ) fr om a catego rification ( b T , D ). Definition 2.20. L et T b e a b alanc e d c omp osabl e e quational t he ory and let ( b T , D ) b e a c ate gorific ation of T . The gr oup P G ( b T , D ) is gener ate d by { T ( b ρ ) A ( b ρ ) | b ρ ∈ Sing ( T ) } , subje ct to the fun ctoriality, natur ality and c oher enc e r elations fr om Definition 2.17, to gether with the fol lowing r elation: ( ˆ ρ b α s,t ) − 1 = ˆ ρ b α t,s . F ollowing the same line of reaso ning as in the pro of of The o rem 2.1 9, we obtain the following relationship b etw een P G ( b T , D ) a nd Struct G ( T ). Theorem 2. 21. L et T b e a b alanc e d, c omp osable e quational t he ory and let ( b T , D ) b e a c ate gori fic ation of T . The fol lowing map is an epimo rphism of gr oups and it is an isomorphi sm if and only if ( b T , D ) is c oher ent: P G ( b T , D ) Θ − → Struct G ( T ) ˆ ρ c α 1 s 1 ,t 1 · . . . · ˆ ρ c α k s k ,t k 7− → ρ α 1 s 1 ,t 1 · . . . · ρ α k s k ,t k  Example 2.22. In Example 2.13, we obtaine d a c oher ent c ate gorific ation of the the ory of semigr oups, S , c onsisting of Mac L ane’s p entagon axiom. It fol lows fr om The or em 2.21 that we c an c onstruct a pr esentation for Struct G ( S ) . We saw in Example 2.9 that Struct G ( S ) is isomorphic to Thompson ’s gr oup F and we ther eby obtain a pr esentation for F . Similarly, we obtain a pr esentation of Thompson ’s gr oup V using the p entagon and hexagon c oher enc e axioms fr om Example 2.13. The r esulting pr esentations ar e the same as those c onstruct e d by Dehornoy [Deh05] . In the following section, we describe gener alisations of Thompson’s groups F and V due to Higman [Hig 7 4] and Brown [Bro87]. 3. The groups F n, 1 and G n, 1 In the previous section, we hav e se en that Thompson’s gr oups F a nd V a rise as structure g roups o f cer tain balanc e d equa tional theor ies and we hav e subseq uent ly obtained presentations for these groups via coherent presentations of their asso - ciated c a tegorica l theories. In this section, w e intro duce generalisations of these 10 JONA THAN A. COHEN groups due to Brown [Br o87] and Higma n [Hig 7 4], which we call F n, 1 and G n, 1 , re- sp ectively . In the following sections, we shall see how the afor ementioned pr o cess of constructing presentations for F and V generalises to this br oader class of groups . There are several paths to defining the groups F n, 1 and G n, 1 , a ll o f which relate to the fact that each o f these g roups arises as a subgro up of the automorphis m group of a Can tor set. O f the m yriad of definitions a v ailable, we cho ose to fo llow the description of Brown [Bro87], which utilise s c e rtain equiv alence clas ses of pair s of finite ro o ted trees. Definition 3.1 (T ree ) . The set of n -ary tr e es is define d inductively as fol lows: • The gr aph c onsisting solely of a single vertex is an n -ary tre e. • If T 1 , . . . , T n ar e n -ary tr e es t hen the fol lowing is also an n -ary tr e e: · T 1 T 2 . . . T n } } } } } } } } }        : : : : : : : : The r o ot of a n n -ary tree is the unique vertex o f v alence 0 or n − 1. The le aves of a ro o ted tr ee T are the vertices of v alence 0 o r 1 and we deno te this set by ℓ ( T ). Definition 3.2 (Expansio n) . A simple expansion of an n -ary tr e e T is the tr e e obtaine d by r eplacing a le af v of T with the fol lowing: v α 1 ( v ) α 2 ( v ) . . . α n ( v ) x x x x x x x x x       : : : : : : : An expansion of an n -ary tr e e is a tr e e obtaine d by making fi nitely many su c c esive simple exp ansions. Given tw o trees T 1 and T 2 having a commo n expa nsion S , we s ay that S is a minimal common expa nsion if any other expa nsion S ′ of T 1 and T 2 is an expa nsion of S . Lemma 3.3 (Higman [Hig74]) . Any two fi nite n -ary t r e es have a minimal c ommon exp ansion.  The underlying sets o f the groups F n, 1 and G n, 1 consist of certa in fo r mal e xpres- sions called tr e e diagr ams . Definition 3.4 (T re e diagram) . An n -a ry tree diagram is a triple ( T 1 , T 2 , σ ) , wher e T 1 and T 2 ar e n -ary tr e es having the same num b er of le aves and σ is a bije ction ℓ ( T 1 ) → ℓ ( T 2 ) . As in the case of trees, we may talk ab out expansions of tree dia grams. Definition 3.5. A simple expa ns ion of an n - ary tr e e diagr am ( T 1 , T 2 , σ ) is an n -ary tr e e diagr am ( T ′ 1 , T ′ 2 , σ ′ ) obtaine d by t he fol lowing pr o c e dur e: • T ′ 1 is a s imple exp ansion of T 1 along the le af l . • T ′ 2 is the simple exp ansion of T 2 along the le af σ ( l ) . • σ ′ is the bije ction ℓ ( T ′ 1 ) → ℓ ( T ′ 2 ) define d by sett ing σ ′ ( k ) = σ ( k ) for k ∈ ℓ ( T 1 ) \ { l } and σ ′ ( α i ( l )) = α i ( σ ( l )) . An expansion of an n -ary t r e e diagr am ( T 1 , T 2 , σ ) is any n -ary tr e e diagr am obtaine d by m aking finitely m any suc c esive simple exp ansions of ( T 1 , T 2 , σ ) . COHERENT PRE SENT A TIONS OF STRUCTURE MONOIDS AND THE HIGMAN-THOMPSON GROUPS 11 Let ∼ b e the equiv alence rela tion on the set of n -ary tree diagra ms obtained by setting ( T 1 , T 2 , σ ) ∼ ( T ′ 1 , T ′ 2 , σ ′ ) whenever ( T 1 , T 2 , σ ) and ( T ′ 1 , T ′ 2 , σ ′ ) p oss ess a common expansion. Let [( T 1 , T 2 , σ )] denote the equiv alence class of ( T 1 , T 2 , σ ) mo dulo ∼ . W e call [( T 1 , T 2 , σ )] a n n -ary tr e e symb ol . Definition 3.6. F or n ≥ 2 , we s et G n, 1 to b e t he gr oup whose underlying set is the c ol le ction of n -ary tre e symb ols, to gether with the fol lowing gr oup struct ur e: • Given two n -ary tre e symb ols [( T 1 , T , σ )] and [( T ′ , T 2 , σ ′ ] , it fol lows fr om L emma 3.3 t hat we may assume that T = T ′ . We define their pr o duct to b e [( T 1 , T , σ )][( T , T 2 , σ ′ )] = [( T 1 , T 2 , σ · σ ′ )] . • The inverse of [( T 1 , T 2 , σ )] is [( T 2 , T 1 , σ − 1 )] . • The un it element is [( T , T , i d )] . It follows from the definitions that an y n - ary tree is an expansion of the tree consisting solely of a s ingle vertex. Thus, the leaves o f an n -a ry tree may b e seen a s a s ubs et of the free mono id on { 1 , . . . , n } . Therefor e, we may order the leaves of the tree lexico g raphically , which is equiv alent to order ing the leav es left-to-rig ht when drawn on a pag e. W e say that an n -a ry tree symbol [( T 1 , T 2 , σ )] is or der-pr eservi ng if σ is a n iso morphism o f ordere d sets; that is, if σ preserves this order ing. Definition 3. 7 . F or n ≥ 2 , we set F n, 1 to b e the sub gr oup of G n, 1 c onsisting of the or der-pr eservi ng n -ary tre e symb ols. The gr o ups F n, 1 and G n, 1 generalise Thompso n’s original g roups F and V , since we hav e F 2 , 1 ∼ = F and G 2 , 1 ∼ = V . They a lso shar e se veral of the in teresting prop- erties of F and V as survey ed in [Sco92]. In the following section, we shall realise F n, 1 as the str ucture gro up o f higher-o rder a sso ciativity and G n, 1 as the str ucture group o f higher or de r ass o ciativity a nd commutativit y . 4. F n, 1 and G n, 1 as str ucture groups Our goal in this section is to realise F n, 1 and G n, 1 as structure groups. Since bo th of these gro ups are built using maps b etw een n -ary trees , we ta ke our set of function symbols to b e F := {⊗} , where ⊗ is an n -ary function symbol. F or a set o f v ariables V , there is an o bvious bijection b etw een F F ( V ) a nd the set of n -ar y tree s whose leaves a re lab elled b y members of V . W e deno te the a bsolutely free term algebra generated by {⊗} on the set V b y F ⊗ ( V ) and we denote the free monoid generated by V by V ∗ . Our basic strategy is to first re alise F n, 1 as a structure g roup by constructing an equational theo r y E such that [ E ] equates any tw o terms t 1 , t 2 ∈ F F ( V ) that co nt a in precisely the same v a riables in the same or der and suc h that no v ariable app ea rs more than onc e in either t 1 or t 2 . In the binary case , there is a n obvious candida te for E : asso ciativity . So, E needs to b e an a nalogue of asso cia tiv ity fo r n > 2. Once we hav e this rea lisation o f F n, 1 we need only add the ability to arbitr arily pe r mut e v ariables in o rder to obtain a realis ation of G n, 1 as a structure gro up. 4.1. Catalan Alge bras and F n, 1 . Asso ciativity of a binary function symbo l is sufficient to establish that an y tw o brack etings of the same string a re equal. The wa y in which one establishes this fact is to show that any bracketing o f a string is equal to the left mo st brack eting. So, for a n n -ary function symbol to be a sso ciative, we need equatio ns which imply that any bra ck eting o f a term is eq uiv alen t to the left most o ne. In order to simplify notation, for int eg ers i ≤ j , we use the symbol x j i to deno te the list x i , x i +1 , . . . , x j . If i > j , then x j i is the empty list. 12 JONA THAN A. COHEN Definition 4. 1 ( n -Catala n algebr as) . F or n ≥ 2 , the the ory of n -Catalan algebr as c onsists of an n -ary function s ymb ol ⊗ to gether with the fol lowing e quations, wher e 0 < i < n : ⊗ ( x i 1 , ⊗ ( x i + n i +1 ) , x 2 n − 1 i + n +1 ) = ⊗ ( x i − 1 1 , ⊗ ( x i + n − 1 i ) , x 2 n − 1 i + n ) We denote the the ory of n - Catalan algebr as by C n . The reason for the name of n -catala n alge bras is that the set of all terms having k o ccurr ences o f the symbol ⊗ a nd containing prec isely one v ariable is in bijective corres p o ndence with the set of n -ary tr e es having k in ternal no de s , which has cardinality equal to the ge ne r alised Ca talan n umber 1 ( n − 1) k +1  nk k  , [Sta99]. The rather opaq ue equationa l theory o f n -Catala n a lgebras is rendered somewhat more understandable by viewing the induced equations on the term tr ees which, for n = 3, yields the following: · x 1 x 2 · x 3 x 4 x 5        : : : : : : :        : : : : : : : = · x 1 · x 5 x 2 x 3 x 4        : : : : : : :        : : : : : : : = · · x 4 x 5 x 1 x 2 x 3 : : : : : : :               : : : : : : : Definition 4.2 (Underlying list) . L et t ∈ F ⊗ ( V ) . Th e underlying list of t is the wor d of V ∗ define d inductively by U ( t ) = ( U ( t 1 ) · . . . · U ( t n ) if t = ⊗ ( t 1 , . . . , t n ) t otherwise Definition 4.3 (Left-mos t brack eting) . L et t ∈ F ⊗ ( V ) . If U ( t ) = t 1 · . . . · t n + k ( n − 1) , then the left-most br acketing of t is define d re cursively by lm b ( t n + k ( n − 1) 1 ) = lmb( ⊗ ( t n 1 ) , t n + k ( n − 1) n +1 ) . W e wish to establish tha t any term F ⊗ ( V ) is equal, in F C n ( V ), to its le ft most brack eting. T o this e nd, we define the r ank and the length of a term, which will b e useful again for a r e lated pr oblem in Sectio n 5. Definition 4.4. L et t ∈ F ⊗ ( V ) . Define t he lengt h of t to b e: L ( t ) = ( P n i =1 L ( t i ) if t = ⊗ ( t n 1 ) 1 otherwise . Define the r ank, R ( t ) , of t inductively by setting R ( t ) = 0 if t ∈ V and R ( ⊗ ( t n 1 )) = n X i =1 R ( t i ) + n X i =2 ( i − 1) L ( t i ) − n ( n − 1) 2 . Note that R ( t ) = 0 pr e cisely when t = lm b( t ) . W e may now pr o ceed to show that a ny term is equiv alent to its left-mos t br ack- eting. Lemma 4. 5. F or any t ∈ F ⊗ ( V ) , we have t = C n lm b( t ) Pr o of. Let t ∈ F ⊗ ( V ). Let R ( t ) a nd L ( t ) be as in Definition 4.4. W e pro ceed b y double induction o n R ( t ) and L ( t ) to show that t = C n lm b( t ). If L ( t ) = 1 then the statement is trivial. W e als o hav e that R ( t ) = 0 if and o nly if t = lmb( t ). COHERENT PRE SENT A TIONS OF STRUCTURE MONOIDS AND THE HIGMAN-THOMPSON GROUPS 13 Suppo se that L ( t ) > 1 a nd R ( t ) > 0 , so that t = ⊗ ( t n 1 ). Let i b e the gr eatest int ege r with the prop erty that t i / ∈ V . If i = 1, then t = lm b( t ) b y induction on L ( t ). If i > 1, then t i = ⊗ ( u n 1 ) and set t ′ = ⊗ ( t i − 2 1 , ⊗ ( t i − 1 , u n − 1 1 ) , u n , t n i +1 ) . A single applica tion of one of the equations in C n establishes that t = C n t ′ . Since R ( t ) − R ( t ′ ) = P n − 1 i =1 L ( u i ), w e ha ve R ( t ′ ) < R ( t ) a nd the statemen t follows by induction on R ( t ).  In or der to manipula te elements of Struct( C n ) effectiv ely , we introduce the notion of a seed. Definition 4 .6 (Seed) . L et F b e a gr ade d set of function symb ols on some set V and let ρ b e a p art ial function F F ( V ) → F F ( V ) . A seed for ρ is a p air of terms s, t ∈ F F ( V ) such that the gr aph of ρ is e qual t o { ( s ϕ , t ϕ ) | ϕ ∈ [ V , F F ( V )] } . In pa rticularly nice ca ses, we can construct seeds for any op erator in a structure monoid. Lemma 4.7 (Dehornoy [Deh00]) . L et T b e a b alanc e d e quational the ory that c on- tains pr e cisely one function symb ol. Th en, e ach op er ator ρ ∈ Struct( T ) admits a se e d. It follo ws from Lemma 3 .3 tha t C n is compo sable and w e ma y , therefore, for m the group Struct G ( C n ). In order to facilitate the passage fro m members of Struc t G ( C n ), to member s of F n, 1 , we introduce the tree gener ated by a term. Definition 4.8. F or a t erm t ∈ F ⊗ ( V ) , let T ( t ) denote t he n -ary t r e e obtaine d via the fol lowing c onst ru ction: • If t = ⊗ ( t 1 , . . . , t n ) , then T ( t ) is e qual to: · T ( t 1 ) T ( t 2 ) . . . T ( t n ) x x x x x x x x x x       : : : : : : : • Otherwise, T ( t ) = · Theorem 4. 9. Struc t G ( C n ) ∼ = F n, 1 . Pr o of. W e denote the seed of ρ ∈ Struct G ( C n ), which exists b y Lemma 4.7, by ( s ρ , t ρ ). W e claim that the following map is an iso mo rphism: Struct G ( C n ) θ − → F n, 1 ρ 7− → [( T ( s ρ ) , T ( t ρ ) , id )] It is ro utine to see that Θ is a homomorphism. Suppose that ρ, ρ ′ ∈ Struct G ( C n ) and that Θ( ρ ) = Θ ( ρ ′ ). It follows that ρ and ρ ′ hav e the same se e d, so ρ = ρ ′ and Θ is faithful. By L e mma 2.6, in order to esta blis h that Θ is surjective, we need only show that t 1 = C n t 2 whenever t 1 , t 2 ∈ F ⊗ ( V ) a nd U ( t 1 ) = U ( t 2 ). By Lemma 4 .5, we hav e t 1 = C n lm b( t 1 ) = C n lm b( t 2 ) = C n t 2 , s o Θ is surjective and, hence, an isomorphism.  14 JONA THAN A. COHEN 4.2. Symmetric Catalan Algebras and G n, 1 . W e s aw in Section 3 that the leav es of a tree may b e ordere d b y the lexicogr aphic order ing o n their addresses. An n -ary tree symbol [( T 1 , T 2 , σ )] may thereby b e viewed as a pair of tree dia g rams, together with a p ermutation of the leav es of T 1 . Thus, in order to obtain an equational theory who se structure gr oup is G n, 1 we need to a dd the ability to arbitrar y p ermute v ariables to Ca talan algebr as. Recalling that the s ymmetric group is generated b y tra nsp o sitions of adjacent e lement s, w e a re led to the following definition. Definition 4.10 (Symmetric n - C a talan Algb e ras) . The the ory of symmetric n - c atalan algebr as extends that of n -c atalan algebr as with the fol lowing e quations, wher e 1 ≤ i < n : ⊗ ( x i − 1 1 , x i , x i +1 , x n i +2 ) = ⊗ ( x i − 1 1 , x i +1 , x i , x n i +2 ) . We denote the the ory of symmet ric n -c atalan algebr as by S C n . Symmetric n -ca ta lan algebra s essentially a dd an ac tio n of the symmetric g r oup on the indices of ⊗ . In gener al, this is sufficient to induce an action of a symmetric group on the v aria bles o f an y term in F ⊗ ( V ). I n the binary case, we re cov er the definition of commutativ e semigroups . Theorem 4. 11. Struct G ( S C n ) ∼ = G n, 1 . Pr o of. F or ρ ∈ Struct G ( S C n ), let ( s ρ , t ρ ) represent its seed, which exis ts by Lemma 4.7. Since S C n is linear , s ρ and t ρ are linear a nd supp( s ρ ) = supp( t ρ ). Let π ( ρ ) b e the p er m utation o f supp( s ρ ) induced by the p ermutation U ( s ρ ) → U ( t ρ ). Consider the following map: Struct G ( C n ) θ − → G n, 1 ρ 7− → [( T ( s ρ ) , T ( t ρ ) , π ( ρ ))] A s imilar argument to the pro of of Theorem 4.9 establishes that Θ is an isomor- phism.  W e no w know that F n, 1 and G n, 1 are the structure groups o f catalan algebras and of symmetric cata lan algebra s, res pe ctively . W e also know that if we ca n construct coherent catego r ifications of these algebras, then we can apply Theorem 2 .21 to obtain presen tations of these gro ups. In the following se c tio n, we set about the task of constr ucting a coher ent ca tegorifica tion of catalan alg ebras. 5. Ca t alan ca tegories and F n, 1 In order to o btain a pres ent a tio n fo r Struct G ( C n ) and, hence, for F n, 1 along the lines of that pro vided by Dehornoy for F [Deh05], w e need to obtain a coherent categorific a tion of C n . The immediate problem is discer ning a se t o f diagra ms whose commutativit y imply the comm utativity of all diagrams in F c C n ( b V ). As w e shall se e in this sectio n, the following definition suffices for this purp ose. While the coherence axioms that w e ha ve chosen may seem slightly cry ptic, the reas on for their choice will become appar ent in the pro o f that the resulting categ o rification is coherent. W e shall ma ke frequent use of the following useful shorthand: F or 1 ≤ i ≤ n and a mor phism ρ : t i → t ′ i , we set ⊗ i ( ρ ) = ⊗ (1 t 1 , . . . , 1 t i − 1 , ρ, 1 t i +1 , . . . , 1 t n ) . Definition 5. 1. A discr ete n -c atalan c ate gory is the c ate gorific ation of C n that c onsists of: • A discr ete c ate gory b V . • A fun ct or ⊗ : b V n → b V . COHERENT PRE SENT A TIONS OF STRUCTURE MONOIDS AND THE HIGMAN-THOMPSON GROUPS 15 • F or 1 ≤ i < n , a natur al isomorphism α i with t he fol lowing c omp onents: α i ( x 2 n − 1 1 ) : ⊗ ( x i 1 , ⊗ ( x i + n i +1 ) , x 2 n − 1 i + n +1 ) → ⊗ ( x i − 1 1 , ⊗ ( x i + n − 1 i ) , x 2 n − 1 i + n ) Pentagon axiom: F or 1 ≤ i ≤ n − 1 , the fo l lowing diagr am c ommutes, wher e X = x i − 1 1 and Z = z n − i − 1 1 : ⊗ ( X , y 1 , ⊗ ( y n 2 , ⊗ ( y 2 n n +1 )) , Z ) ⊗ ( X , ⊗ ( y n 1 ) , ⊗ ( y 2 n n +1 ) , Z ) ⊗ ( X , y 1 , ⊗ ( y n − 1 i , ⊗ ( y 2 n − 1 n ) , y 2 n ) , Z ) ⊗ ( X , ⊗ ( ⊗ ( y n 1 ) , y 2 n − 1 n +1 ) , y 2 n , Z ) ⊗ ( X , ⊗ ( y n − 1 1 , ⊗ ( y 2 n − 1 n )) , y 2 n , Z ) α i | | ⊗ i +1 ( α n − 1 ) α i   α i   ⊗ i ( α n − 1 · ... · α 1 ) g g A djac ent asso ciativity axiom: F or 1 ≤ i ≤ n − 2 , the fol lowing diagr am c ommutes, wher e X = x i − 1 1 and Z = z n − i − 2 1 : ⊗ ( X , y 1 , ⊗ ( y n +1 2 ) , ⊗ ( y 2 n +1 n +2 ) , Z ) ⊗ ( X , ⊗ ( y n 1 ) , y n +1 , ⊗ ( y 2 n +1 n +2 ) , Z ) ⊗ ( X , y 1 , ⊗ ( ⊗ ( y n +1 2 ) , y 2 n n +2 ) , y 2 n +1 , Z ) ⊗ ( X , ⊗ ( y n 1 ) , ⊗ ( y 2 n − 1 n +1 ) , y 2 n +1 2 n , Z ) ⊗ ( X , ⊗ ( y 1 , ⊗ ( y n +1 2 ) , y 2 n − 1 n +2 ) , y 2 n − 1 2 n , Z ) ⊗ ( X , ⊗ ( ⊗ ( y n 1 ) , y 2 n − 1 n +1 ) , y 2 n +1 2 n , Z ) α i ~ ~ α i +1   α i +1   α i   α i   ⊗ i ( α 1 ) p p We denote the the ory of discr ete n -c atalan c ate gories by C n and the fr e e C n c ate gory on b V by F C n ( b V ) . In the case where n = 2, the p entagon axiom reduces to Ma c Lane’s pentagon axiom for monoidal catego ries from E xample 2.13 and the adjacent asso cia tivit y axiom is empty , so w e recover the usual definition of a coherently a sso ciative bi- functor. Definition 5.2 (Positive/Negative) . A singular morphism of F C n ( b V ) that c ontains an instanc e of α i is c al le d pos itive and one that c ontains an instanc e of α − 1 i is c al le d negative . A morphism in F C n ( b V ) is c al le d pos itive if it is an identity or a c omp osite of p ositive morphisms and negative if it is a c omp osite of ne gative morphisms. 16 JONA THAN A. COHEN It follows fro m the pro of of Lemma 4.5 that there is a lwa ys a p ositive mor phism t → lm b( t ) in F C n ( b V ). In order to show tha t C n is a coherent categorifica tio n of C n , we need to show that any diag ram built out of the sing ular and ident ity morphisms of C n commutes. As our first step tow ards this goa l, we show that there is a unique po sitive morphism t → lmb( t ). Lemma 5. 3 . L et t ∈ O b( F C n ( b V )) . Ther e is a unique p ositive morphism t → lmb( t ) in F C n ( b V ) . Pr o of. Let t ∈ Ob( F C n ( b V )). It follows from the pr o of of Lemma 4.5 that there is a po sitive mo rphism t → lmb( t ). Suppos e that ϕ, ψ : t → lm b( t ), tha t ϕ = ϕ 1 · ϕ 2 and that ψ = ψ 1 · ψ 2 . By Lemma 2.15, we may as sume that ϕ 1 and ψ 1 are singula r. Let R ( t ) and L ( t ) b e defined as in Definition 4 .4. W e pro ceed by double induction o n R ( t ) and L ( t ) to show that there exis ts an ob ject w in F C n ( b V ) mak ing the following diagram commute: t u v w lmb( t ) (1) (2) (3) ϕ 1 7 7 ψ 1 ' '   F : 0 I I x   ϕ 2   ψ 2 _ _ / / _ _ _ _ _ _ By induction on R ( t ), we hav e that the sub diagr ams lab elled (2) and (3) c o mmut e. So, we need o nly establish the existence and commutativit y of the sub diagra m lab elled (1). If R ( t ) = 0 or L ( t ) = 1, then the statement is trivia l, s o supp o se that R ( t ) > 0 and L ( t ) > 1. If ϕ 1 = ψ 1 , then take w = u = v and s ub diag ram (1) commutes trivially . Suppo se tha t ϕ 1 6 = ψ 1 . Then, either ϕ 1 = ⊗ i ( ϕ ′ 1 ) or ϕ 1 = α i ( t n 1 ) and ther e a re similar p o s sibilities for ψ 1 . W e pro ceed by case a nalysis on the form of ϕ 1 and ψ 1 . Suppo se that ϕ 1 = ⊗ i ( ϕ ′ 1 ) and ψ 1 = ⊗ j ( ψ ′ 1 ). If i = j , then the who le dia gram commutes by induction on L ( t ). Supp os e that i 6 = j . Then, without loss of gener- ality , i < j and we may take (1) to b e the following diagr am, which commutes by the functoriality of ⊗ : ⊗ ( t n 1 ) ⊗ ( t i − 1 1 , t ′ i , t n i +1 ) ⊗ ( t j − 1 1 , t ′ j , t n j +1 ) ⊗ ( t i − 1 1 , t ′ i , t j − 1 i +1 , t ′ j , t n j +1 ) ϕ 1   ψ 1   ψ 1 % % ϕ 1 z z Suppo se that ϕ 1 = ⊗ i ( ϕ ′ 1 ) and ψ 1 = α j ( t n 1 ). If i 6 = j , then witho ut loss o f generality i < j a nd t j = ⊗ ( u n 1 ). If i 6 = j − 1, then w e may take (1) to be the COHERENT PRE SENT A TIONS OF STRUCTURE MONOIDS AND THE HIGMAN-THOMPSON GROUPS 17 following square, which commutes by the naturality of α i : ⊗ ( t j − 1 1 , ⊗ ( u n 1 ) , t n j +1 ) ⊗ ( t i − 1 1 , t ′ i , t j − 1 i +1 , ⊗ ( u n 1 ) , t n j +1 ) ⊗ ( t j − 2 1 , ⊗ ( t j − 1 , u n − 1 1 ) , u n , t n j +1 ) ⊗ ( t i − 1 1 , t ′ i , t j − 2 i +1 , ⊗ ( t j − 1 , u n − 1 1 ) , u n , t n j +1 ) ϕ 1 { { ψ 1 # # ψ ′ 1 ( ( ϕ ′ 1 w w If i = j − 1, then we may take (1) to be a similar naturality sq uare. Suppose that i = j . If ϕ ′ 1 = ⊗ k ( ϕ ′′ 1 ), then we ma y take (1) to be a naturality square. If ϕ ′ 1 = α k ( u n 1 ) then we hav e t wo cases. If k 6 = ( n − 1), then w e may take (1 ) to b e a na tur ality square. If k = n − 1, then we may take (1) to b e an instanc e of the pentagon axiom, which commutes by assumption. Finally , we are left with the ca se where ϕ 1 = α i ( t n 1 ) and ψ 1 = α j ( t n 1 ). If | i − j | > 1, then we may take (1) to b e a naturality diagr am. If | i − j | = 1 , then we may take (1) to b e an instance o f the a djacent asso ciativity axiom.  W e now k now that every ob ject in F C n ( b V ) has a unique p ositive morphism to its left-most brack eting. With a little work, w e ca n bo otstra p this result in order to show that there is a unique morphism - positive, negative or otherwise - b etw een any tw o arbitrar y ob jects in F C n ( b V ). Theorem 5. 4. C n is a c oher ent c ate gorific ation of C n . Pr o of. Supp ose that ϕ : s → t is a reduction in F C n ( b V ). By Lemma 2.1 5, ϕ = s ϕ 1 → s 1 ϕ 2 → s 2 → · · · ϕ n − 1 → s n − 1 ϕ n → t, where ea ch ϕ i is s ingular. By Lemma 5.3, each term t ∈ F C n ( b V ) has a unique map N t : t → lm b( t ). W e claim tha t each r ectangle in the following diagr am commutes: s ϕ 1 / / N s   s 1 ϕ 2 / / N s 1   s 2 ϕ 3 / / N s 2   . . . ϕ n − 1 / / s n − 1 ϕ n / / N s n − 1   t N t   lm b( s ) lm b( s 1 ) lm b( s 2 ) . . . lm b( s n − 1 ) lmb( t ) If ϕ i is p ositive, then it follows immediately from Lemma 5 .3 that ϕ i · N s i = N s i − 1 . If ϕ i is negative, then Lemma 5.3 implies that ϕ − 1 i · N s i − 1 = N s i , which implies that ϕ i · N s i = N s i − 1 . Since ea ch rectangle commutes, we hav e ϕ · N t = N s , which implies that ϕ = N s · N − 1 t . Since N s and N t are unique a nd we did not rely on a particular choice of ϕ , we co nclude tha t C n is c o herent.  With Theorem 5.4 in hand, we can obtain a pres entation for F n, 1 , which gener- alises the prese ntation for F g iven in [Deh05 ]. Corollary 5.5. P G ( C n ) ∼ = F n, 1 Pr o of. By Theor e m 5.4 and Theorem 2 .21, we hav e P − G ( C n ) ∼ = Struct G ( C n ). It follows then from Theor em 4 .9 that P G ( C n ) ∼ = F n, 1 .  In the following section, we shall obtain a co herent ca teg orificatio n o f S C n and, thereby , a presentation o f G n, 1 . 18 JONA THAN A. COHEN 6. Symmetric ca t alan ca tegories and G n, 1 Our goal in this section is to construct a co herent catego rification of symmet- ric ca talan alg ebras. The co herence theor e m for catalan categories, Theo rem 5 .4, reduces this problem to ensuring that an y t wo sequences of transpo sitions o f the ob jects app earing in a term realis e the same per mu tatio n. In other words, our cate- gorificatio n needs to so mehow enco de a pr esentation of the symmetric gro up whose generator s cor resp ond to tra nsp ositions of adjacent v ariables. Such a presentation is well known, having b een constructed by Mo ore [Mo o9 6]. This pres ent a tio n has generator s T 1 , . . . , T n − 1 and the following rela tio ns: T 2 i = 1 for 1 ≤ i ≤ n − 1 ( T i T i +1 ) 3 = 1 for 1 ≤ i ≤ n − 2 ( T i T k ) 2 = 1 for 1 ≤ i ≤ k − 2 With this presentation in mind, we may now co nstruct a r easonable categ orification of S C n . Reca ll our shorthand that for 1 ≤ i ≤ n and a mor phism ρ : t i → t ′ i , w e hav e ⊗ i ( ρ ) = ⊗ (1 t 1 , . . . , 1 t i − 1 , ρ, 1 t i +1 , . . . , 1 t n ) . Definition 6.1. F or n ≥ 2 , a discr ete symmetric n -c atala n c ate gory, is a di scr ete n -c atalan c ate gory on the c ate gory b V , to gether with, for 1 ≤ i ≤ n − 1 , a natur al isomorphi sm τ i with c omp onents τ i ( t n 1 ) : ⊗ ( t i − 1 1 , t i , t i +1 , t n i +2 ) → ⊗ ( t i − 1 1 , t i +1 , t i , t n i +2 ) , satisfying the fol lowing axio ms: Involution axiom: F or 1 ≤ i ≤ n − 1 , the fol lowing diagr am c ommu tes: ⊗ ( t n 1 ) ⊗ ( t n 1 ) ⊗ ( t i − 1 1 , t i +1 , t i , t n i +2 ) 1   τ i ( ( Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q τ i o o Comp atibility axiom: F or 2 ≤ i ≤ n and 1 ≤ j ≤ n − 2 , t he fol lowing diagr am c ommutes, wher e W = w i 1 and Z = z n − i 1 : ⊗ ( W , x, ⊗ ( y n 1 ) , Z ) ⊗ ( W , ⊗ ( x, y n − 1 1 ) , y n , Z ) ⊗ ( W , x, ⊗ ( y j − 1 1 , y j +1 , y j , y n j +2 ) , Z ) ⊗ ( W , ⊗ ( x, y j − 1 1 , y j +1 , y j , y n − 1 j +2 ) , y n , Z ) α i − 1 ~ ~ ⊗ i ( τ j ) ! ! ⊗ i − 1 ( τ j +1 ) & & α i − 1 w w COHERENT PRE SENT A TIONS OF STRUCTURE MONOIDS AND THE HIGMAN-THOMPSON GROUPS 19 3 -cycle axiom: F or 1 ≤ i ≤ n − 2 , t he fol lowing diagr am c ommutes: ⊗ ( t n 1 ) ⊗ ( t i − 1 1 , t i +1 , t i , t n i +2 ) ⊗ ( t i 1 , t i +2 , t i +1 , t n i +3 ) ⊗ ( t i − 1 1 , t i +1 , t i +2 , t i , t n i +3 ) ⊗ ( t i − 1 1 , t i +2 , t i , t i +1 , t n i +3 ) ⊗ ( t i − 1 1 , t i +2 , t i +1 , t i , t n i +3 ) τ i ~ ~ τ i +1 τ i +1   τ i   τ i + + τ i +1 w w Hexagon ax iom: F or 1 ≤ i ≤ n − 1 , the fol lowing diagr am c ommutes, wher e W = w i − 1 1 and Z = z n − i − 1 1 : ⊗ ( W , ⊗ ( x n 1 ) , y , Z ) ⊗ ( W , y , ⊗ ( x n 1 ) , Z ) ⊗ ( W , x 1 , ⊗ ( x n 2 , y ) , Z ) ⊗ ( W , ⊗ ( y , x n − 1 1 ) , x n , Z ) ⊗ ( W , x 1 , ⊗ ( y , x n 2 ) , Z ) ⊗ ( W , ⊗ ( x 1 , y , x n − 1 2 ) , x n , Z ) τ i   α − 1 i α i   ⊗ i +1 ( τ n − 1 · ... · τ 1 )   ⊗ i ( τ 1 ) + + α i w w We denote the the ory of discr ete symm et ric n -c atala n c ate gories by SC n and the fr e e SC n -c ate gory on b V by F SC n ( b V ) . The hexagon axiom ensures that we ma y re place a transpo sition of the form τ i ( t i − 1 1 , ⊗ ( u n 1 ) , t n − 1 i ) with a se quence of tra nsp ositions inv olving o nly the terms t n − 1 1 and u n 1 . One might posit the c o mmut ativity of a diagr a m that serves the same purpose for a morphism of the form τ i ( t i 1 , ⊗ ( u n 1 ) , t n − 1 i +1 ). Do ing s o le ads to the dual hexagon diagr am , which has the following form, where 2 ≤ i ≤ n a nd 20 JONA THAN A. COHEN W = w i − 2 1 and Z = z n − i 1 : ⊗ ( W , x, ⊗ ( y n 1 ) , Z ) ⊗ ( W , ⊗ ( y n 1 ) , x, Z ) ⊗ ( W , ⊗ ( x, y n − 1 1 ) , y n , Z ) ⊗ ( W , y 1 , ⊗ ( y n 2 , x ) , Z ) ⊗ ( W , ⊗ ( y n − 1 1 , x ) , y n , Z ) ⊗ ( W , y 1 , ⊗ ( y n − 1 2 , x, y n ) , Z ) τ i   α i α − 1 i   ⊗ i ( τ 1 · ... · τ n − 1 )   ⊗ i +1 ( τ n − 1 ) + + α − 1 i w w Lemma 6. 2. The dual hexago n di agr am c ommutes in F SC n ( b V ) . Pr o of. By the inv olutio n a xiom, we hav e τ i ( t i 1 , ⊗ ( u n 1 ) , t n − 1 i +1 ) = [ τ i ( t i − 1 1 , ⊗ ( u n 1 ) , t n − 1 i )] − 1 . Using the hexago n axiom and ig noring comp onent lab els, we have: τ i = τ − 1 i = ( α − 1 i · ⊗ i +1 ( τ n − 1 · . . . · τ 1 ) · α i · ⊗ i ( τ 1 ) · α − 1 i ) − 1 So, we hav e: τ i · α − 1 i · ⊗ i +1 ( τ n − 1 ) = α i · ⊗ i ( τ 1 ) · α − 1 i · ⊗ i +1 ( τ 1 , . . . , τ n − 2 ) = α i · ⊗ i ( τ 1 ) · α − 1 i · ⊗ i +1 ( τ 1 ) · . . . · ⊗ i +1 ( τ n − 2 ) (1) F rom the compatibility a xiom, we hav e ⊗ i +1 ( τ j ) = α i · ⊗ i ( τ j +1 ) · α − 1 i . This implies: (1) = α i · ⊗ i ( τ 1 ) · . . . · ⊗ i ( τ n − 1 ) · α − 1 i = α i · ⊗ i ( τ 1 · . . . · τ n − 1 ) · α − 1 i Therefore, the dual hexa gon diag ram co mmu tes in F SC n ( b V ).  In the n = 2 ca se, the axiomatisa tion o f SC n reduces to the theory of a c o herently asso ciative a nd co mmutative bifunctor given in Exa mple 2 .1 3. The main r e sult of this section establishes that SC n is a s uitable gener a lisation of this case. Theorem 6. 3. SC n is a c oher ent c ate gorific ation of S C n . Pr o of. By Theo rem 5.4, we may assume that all of the a sso ciativity maps ar e strict equalities. Th us, an ob ject of F SC n ( b V ) may b e re pr esented as ⊗ ( t m 1 ), where each t i is an ob ject in b V and m = n + k ( n − 1), for some k ≥ 0. Lemma 6.2 and the hexagon axio m imply that it suffices to consider transp ositions of adjacent v a riables that are ob jects of b V . So, for a given ob ject t := ⊗ ( t m 1 ), we need o nly consider the m − 1 induced tra nsp osition natural iso morphisms T i ( t m 1 ) : ⊗ ( t i − 1 1 , t i , t i +1 , t m i +2 ) → ⊗ ( t i − 1 1 , t i +1 , t i , t m i +2 ) . In order to es tablish co herence, w e ha ve to sho w that every p ermutation of t m 1 is unique. That is, we hav e to show that the induced transp ositio n maps satisfy the defining relatio ns for the sy mmetr ic g roup of order m . COHERENT PRE SENT A TIONS OF STRUCTURE MONOIDS AND THE HIGMAN-THOMPSON GROUPS 21 The compatibility axiom implies that each T i is unique. By the naturality of the maps T i , we hav e T i · T k = T k · T i for all 1 ≤ i ≤ k − 2. The inv olution a xiom implies that T 2 i = 1. Thus, it o nly remains to establish that ( T i · T i +1 ) 3 = 1. F o r n = 2 , we may use the pro of from Mac Lane [ML63]. Supp ose that n ≥ 3. Since the asso ciativity ma ps ar e taken to b e strict equalities, we may a ssume that t ha s the form ⊗ ( R, ⊗ ( S, t i , t i +1 , t i +2 , U ) , V ), where R, S, U and V are sequences of ob jects of b V . The res ult then follows from the 3-cycle axiom.  With the coherence theorem in hand, we c a n co nstruct a pres e nt atio n of Struct G ( S C n ) and, therefor e, of G n, 1 , which gener alises the presentation for V given in [Deh05]. Corollary 6.4. P G ( SC n ) ∼ = G n, 1 Pr o of. By Theor em 6 .3 and Theorem 2.21, we have P G ( SC n ) ∼ = Struct G ( S C n ) . It follows then from Theor em 4 .11 tha t P G ( C n ) ∼ = G n, 1 .  7. Conclusions and fur ther w ork W e hav e demonstrated, by wa y of Theor em 2.19 and Theorem 2.21, that there is a clo se relations hip b etw een structure monoids and coherent ca tegorica l theories . This relations hip is quite p ow erful, as illustrated by the fact that we were a ble to exploit it in order to o btain new presentations of F n, 1 and G n, 1 . While we only dea lt with inv ertible categoric al structures , it is str a ightforw ar d to e x tend the constructio ns o f Sectio n 2 to structures inv olving a mix of inv ertible and non-inv er tible natur a l transfor mations. Within this setting, it is p os sible to develop an abstract c o herence theor em that applies to a large array of structures and, inter alia, yields pre s entations of a wide v ariety o f structure mo noids. A general coherence theorem alo ng the line s of the pro o f o f Theorem 5.4 is develope d in [CJ]. A more p ow erful, though mo r e difficult to apply , general coher ence theo rem applying mainly to badly b ehaved no n-inv ertible str ucture is developed in [Co h]. The presentations that aris e from c o herence theo rems a r e all infinite, rega r dless of whether o r not the asso ciated algebra ic structure is finitely pre sentable. In particular, this is the case for the pr esentations of F n, 1 and G n, 1 that arise from the coherence theo rems for catalan categ ories and symmetric catalan categories . How ever, these presentations aris e fro m finitely pre sented ca tegorica l structures and so retain so me a mount of finiteness. Conditions on a finitely presented categor ical structure ensuring that it is coherent via finitely many coherence conditions ar e obtained in [Coh]. The r elation b etw een the finite presentabilit y o f a coherent categoric al structure and the finite pres entabilit y of the monoid o r gr o up that a rises from it remains unclear a nd muc h r emains to be done in this directio n. References [Bro87] Kenneth S. Brown. Fi niteness prop erties of groups. In Pr o c e e dings of the Northwestern c onfer enc e on c ohom olo gy of gr oups (Evanston, Il l., 1985) , v olume 44, pages 45–75 , 1987. [CJ] Jonathan A. Cohen and Micha el Johnson. Complete term rewriting theories and cate- gorical coherence problems. In Prep er ation. [Coh] Jonathan A. Cohen. Coherence without unique normal f orms. In prep eration. [Deh93] P atrick Dehornoy . Structural monoids asso ciated to equational v arieties. Pr o c. Amer. Math. So c. , 117(2):293–304, 1993. [Deh00] P atrick Dehornoy . 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Concerning the abstract groups of order k ! and 1 2 k ! holohedri- cally isomorphi c wi th the symmetric and the alternating substitution-groups on k letters. Pr o c. L ondon Math. So c. , 28:357–367 , 1896. [Sco92] E. A. Scott. A tour around finitely presented i nfinite s imple groups. In Algorithm s and classific ation in c ombinatorial gr oup t he ory (Berkeley, CA, 1989) , volume 23 of Math. Sci. R es. Inst. Publ. , pages 83–119. Springer, New Y ork, 1992. [Sta99] Ric hard P . Stanley . Enumer ative c ombinatorics. V ol. 2 , v olume 62 of Cambridge Studies in A dvanc e d Mathematics . Cambridge Universit y Pr ess, Cambridge, 1999. Dep ar tment of Computing, Macquarie University, S ydney NSW 210 9, Australia E-mail addr ess : jonathan.cohen @anu.edu.au