A trace group (monoid) is the quotient of a free group (monoid) by relations of commutation between some pairs of generators. We prove an analog for the trace group of the Möbius inversion formula for the trace monoid (Cartier and Foata, 1969)

Anne Bouillard

Jean Mairesse

Comptes Rendus Mathématique

1/f gener-a, 1969).’unlogue pourtationélisertérêts, lesfois,uenC. R. Acad. Sci. Paris, Ser. I 339 (2004) 899–904http://france.elsevier.com/direct/CRASSComputer ScienceMöbius inversion formula for the trace groupAnne Bouillard, Jean MairesseLIAFA, CNRS – université Paris 7, case 7014, 2, place Jussieu, 75251 Paris cedex 05, FranceReceived 1 January 2004; accepted after revision 15 October 2004Available online 11 November 2004Presented by Maurice NivatAbstractA trace group (monoid) is the quotient of a free group (monoid) by relations of commutation between some pairs oators. We prove an analog for the trace group of the Möbius inversion formula for the trace monoid (Cartier and FoatTo cite this article: A. Bouillard, J. Mairesse, C. R. Acad. Sci. Paris, Ser. I 339 (2004). 2004 Académie des sciences. Published by Elsevier SAS. All rights reserved.RésuméUne formule d’inversion de Möbius pour le groupe de traces. Un groupe (monoïde) de traces est le quotient dgroupe (monoïde) libre par des relations de commutation entre certaines paires de générateurs. On montre un anale groupe de traces de la formule d’inversion de Möbius pour le monoïde de traces (Cartier et Foata, 1969).Pour citer cetarticle : A. Bouillard, J. Mairesse, C. R. Acad. Sci. Paris, Ser. I 339 (2004). 2004 Académie des sciences. Published by Elsevier SAS. All rights reserved.Version française abrégéeUn groupe (monoïde) de traces est le quotient d’un groupe (monoïde) libre par des relations de commuentre certaines paires de générateurs (lettres). Les monoïdes de traces sont classiquement utilisés pour modles exécutions dans les systèmes concurrents, voir [5] et les références qui s’y trouvent. Entre autres ingroupes de traces permettent d’«approximer » les groupes de tresses [16].Un élémentm d’un monoïde de tracesM est uneclique si chaque lettre le composant apparaît au plus uneet si toutes ces lettres commutent. On noteC l’ensemble des cliques deM. Soit |m| la longueur dem ∈ M définieen (3). SoitµM ∈ Z〈M〉 le polynôme défini parµM = ∑c∈C(−1)|c|c. Dans [3, Théorème 2.4], il est prouvé qµM est l’inverse formel dansZ〈〈M〉〉 de la série caractéristique deM, c’est-à-dire que l’on a l’identité (4). OE-mail addresses: bouillard@liafa.jussieu.fr (A. Bouillard), mairesse@liafa.jussieu.fr (J. Mairesse).1631-073X/$ – see front matter 2004 Académie des sciences. Published by Elsevier SAS. All rights reserved.doi:10.1016/j.crma.2004.10.017900 A. Bouillard, J. Mairesse / C. R. Acad. Sci. Paris, Ser. I 339 (2004) 899–904tude-é-[10,12]yofnt[11],eries.someons.polyno-idons andöbiusction ofeferencesing eventsames likeate’1.appelle cette identité la ‘formule d’inversion de Möbius pour le monoïde de traces’. Elle est au cœur de l’écombinatoire du monoïde de traces [4,5].On obtient dans cette Note une ‘formule d’inversion de Möbius pour le groupe de traces’. SoitF un groupe detraces sur l’alphabetΣ ∪ Σ (ā ∈ Σ est l’inverse dea ∈ Σ). SoitM le monoïde de traces surΣ ∪ Σ dont la présentation de monoïde est obtenue à partir de la présentation de monoïde deF n supprimant les relationsaā = āa = 1,a ∈ Σ . Soit φ l’injection canonique deF dansM. (Plus précisément,φ(t), t ∈ F, est la projection dansM d’unreprésentant de longueur minimale det dans(Σ ∪ Σ)∗.) Soit D = T (C) où T :M → P(M) est la fonction dé-finie parT (a) = a(āa)∗(1 + ā), a ∈ Σ ∪ Σ , et étendue par morphisme. On démontre que(∑d∈D(−1)|d |d) estl’inverse formel dansZ〈〈M〉〉 de la série caractéristique deφ(F), c’est-à-dire que l’on a l’identité (7). Une consquence de cette identité est que la série de croissance deF s’exprime comme :∑t∈F x |t | =∑t∈M x |t |/(1+ x)|t | =(∑c∈C(−1)|c|x |c|/(1+ x)|c|)−1. Ceci permet d’exporter vers le groupe de traces les résultats prouvés danspour le monoïde de traces.1. IntroductionA decomposition of an elementm of a monoidM is a n-uple (m1, . . . ,mn) satisfyingm1 · · ·mn = m withmi ∈ M \ {1} (where 1 is the unit of the monoid). The decomposition iseven (odd ) if n is even (odd). Assume anelement ofM has a finite number of decompositions and set(µM |m) to be the number of even decompositionsm minus the number of odd decompositions. ViewµM as a formal series ofZ〈〈M〉〉 and call it theMöbius seriesof M. A one line computation shows thatµM is the formal inverse of the characteristic series ofM: in Z〈〈M〉〉,(∑m∈M m) · µM = 1. This identity is called aMöbius inversion formula, see Cartier and Foata [3], Lalleme[13], or Rota [15] in a different setting. The classical Möbius inversion principle in number theory (seeChapter XVI) is a special instance of the identity. It is generally difficult to effectively compute the Möbius sA trace group (monoid) is the quotient of a free group (monoid) by relations of commutation betweenpairs of generators (letters). Observe that in a trace monoid, the traces all have a finite number of decompositiA classical result of Cartier and Foata [3, Theorem 2.4] states that the Möbius series of a trace monoid is amial which is explicitly computable, see§3. This is the starting point of the combinatorial study of the trace mono[4,5]. In a trace group as in any non-trivial group, the elements do not have a finite number of decompositithe Möbius series is not defined. However, we prove in this note an identity which has the flavor of a Minversion formula, see Theorem 4.1. As a by-product we express the growth series of a trace group in funthe growth series of a canonically associated trace monoid, see Corollary 4.2.Trace monoids are often used to model the occurrence of events in concurrent systems, see [5] and the rtherein. Roughly speaking, a letter corresponds to an event and two letters commute when the correspondcan occur simultaneously. Trace groups have been studied from several viewpoints (and under various ngraph groups or right angled Artin groups) [6,8,16]. An important motivation is that trace groups can ‘approximbraid groups [16].2. Trace monoids and trace groupsLet Σ be a non-empty finite set and denote byΣ∗ the free monoid overΣ . The empty word is denoted byThe monoid presented byΣ andR ⊂ Σ∗ × Σ∗ is denoted by〈Σ | u = v, (u, v) ∈ R〉. Let I ⊂ Σ × Σ be ananti-reflexive and symmetric relation, called anindependence (or commutation) relation. Thetrace monoid (or freepartially commutative monoid) M(Σ, I) is defined by the monoid presentationM(Σ, I) = 〈Σ | ab = ba, ∀(a, b) ∈ I 〉. (1)The elements ofM(Σ, I) are calledtraces.A. Bouillard, J. Mairesse / C. R. Acad. Sci. Paris, Ser. I 339 (2004) 899–904 901dthetivece byn aowhe tracetheacehdyt trace4.7]Let Σ be a copy ofΣ , that is a set which is in bijection withΣ and disjoint fromΣ ; to each lettera ∈ Σcorresponds the letter̄a in Σ . SetΣ̃ = Σ ∪ Σ . We extend the bar notation tõΣ , by settingā = a. Denote byF(Σ) the free group overΣ , with Σ as the set of inverses of the generators. LetF(Σ, I) be defined by the monoipresentationF(Σ, I) = 〈Σ̃ | aā = āa = 1, ∀a ∈ Σ; ab = ba, ∀(a, b) ∈ I 〉. (2)It is easily checked thatF(Σ, I) is a group, called thetrace group (or free partially commutative group). A grouppresentation of the trace group isF(Σ, I) = 〈Σ | ab = ba, ∀(a, b) ∈ I 〉. The elements ofF(Σ, I) are also calledtraces.WhenI = ∅, the corresponding trace monoid is the free monoidΣ∗ and the corresponding trace group isfree groupF(Σ). WhenI = Σ × Σ \ {(a, a), a ∈ Σ}, the corresponding trace monoid is the free commutamonoidNΣ and the corresponding trace group is the free commutative groupZΣ .Defineι : Σ̃ → Σ by: ∀a ∈ Σ, ι(a) = ι(ā) = a. The relationĨ ⊂ Σ̃ × Σ̃ is defined by(u, v) ∈ Ĩ ⇐⇒ (ι(u), ι(v)) ∈ I.The diagram below, where the applications are the canonical surjective morphisms, is commutative.Σ̃∗πFπMM(Σ̃, Ĩ )ϕF(Σ) F(Σ, I)The notations are shortened toπ andϕ when there is no possible confusion. It is convenient to denote a traany of its word representatives, that is to identifyw andπ(w). For instance, 1 denotes the empty trace. Givewordw ∈ Σ∗ and a lettera ∈ Σ , denote by|w|a ∈ N the number of occurrences of the letter in the word. We ngive several definitions which are stated in order to hold both in a trace monoid and in a trace group. For tmonoid only, the same definitions take a simpler form. LetS be the set of generators (including the inverses forgroup). Thelength (with respect toS) of a tracet is:|t| = min{k | t = g1 · · ·gk, gi ∈ S}. (3)Observe that|t| = min{|x|, x ∈ π−1(t)}. Given a tracet and a lettera, define|t|a = min{|x|a, x ∈ π−1(t)},and define thealphabet of t by alph(t) = {a ∈ S | |t|a > 0}. A trace t is a clique if: ∀a ∈ alph(t), |t|a = 1 and∀a, b ∈ alph(t), ab = ba. We denote the set of cliques byC. Observe that 1∈ C.We visualize traces using Viennot’s representation as heaps of pieces [17,16]. In the trace monoid, a trcorresponds to aheap. In the trace group, a trace corresponds to acolored heap (letters ofΣ are associated witlight gray pieces, letters ofΣ with dark gray pieces, and, in the colored heap, consecutive pieces of typea andācancel each other).Example 1. The Basic Example to be used as illustration throughout the paper consists ofΣ = {a, b, c} andI = {(a, b), (b, a)}. We have represented in Fig. 1, the traces inM(Σ̃, Ĩ ) andF(Σ, I) corresponding to the woru = aab̄c̄cāb̄. We have|πM(u)| = 7 and|πF(u)| = 3.The heap representation enables to define theCartier–Foata normal form for traces by reading a trace ‘slice bslice’ in the associated heap [3]. A corollary is that the word problem is solvable in polynomial time and thagroups (monoids) are automatic in the sense of [9]. Another consequence is the following, see [7, Proposition 2.for a formal proof. Givenu,v ∈ Σ̃∗ such thatπF(u) = πF(v) and |u| = |v| = |πF(u)|, then we haveπM(u) =πM(v). In other words, a trace ofF(Σ, I) admits a unique representative of minimal length inM(Σ̃, Ĩ ). We denote902 A. Bouillard, J. Mairesse / C. R. Acad. Sci. Paris, Ser. I 339 (2004) 899–904in4)Fig. 1. A heap inM(Σ̃, Ĩ ) (left) and a colored heap inF(Σ, I ) (right).by φ :F(Σ, I) → M(Σ̃, Ĩ ) the corresponding map. Using the language of heaps, ift is a colored heap thenφ(t) isthe corresponding (non-colored) heapobtained by forgetting the colors.Let M be a trace monoid or a trace group. Thegrowth series of M is the series GrM ∈ Z[[x]] defined byGrM = ∑m∈M x |m| = ∑n∈N #{m ∈ M | |m| = n} xn. Setan = #{m ∈ M | |m| = n}. The growth rate of M isρM = limn a1/nn  1 (the limit exists by sub-additivity of the sequence(logan)n).3. Möbius formula for the trace monoidThe results recalled here are classical and dueto Cartier and Foata [3, Theorem 2.4]. LetM(Σ, I) be a tracemonoid. Denote byZ〈〈M(Σ, I)〉〉 the ring of formal power series overM(Σ, I) with coefficients inZ. Thisring is cancellative, in particular a right or left inverse is an inverse. WhenM(Σ, I) = {x}∗, we writeZ[[x]] forZ〈〈M(Σ, I)〉〉. The following identity holds inZ〈〈M(Σ, I)〉〉:( ∑t∈M(Σ,I )t)·(∑c∈C(−1)|c|c)= 1. (4)Using the terminology of the Introduction, the Möbius series ofM(Σ, I) is µM(Σ,I ) = ∑c∈C(−1)|c|c. DefineµM(Σ,I )(x) ∈ Z[[x]] by µM(Σ,I )(x) = ∑c∈C(−1)|c|x |c|. An easy consequence of (4) is the following identityZ[[x]]:GrM(Σ, I) · µM(Σ,I )(x) = 1. (5)So, the growth series of the trace monoid is the formal inverse ofµM(Σ,I )(x). The objective is to get analogs of (and (5) for trace groups.4. Möbius formula for the trace groupDefine the set ofalternate traces as the subsetD of M(Σ̃, Ĩ ) given by the regular expressionD =∑c∈C[ ∏x∈alph(c)x(x̄x)∗(1+ x̄)], (6)Fig. 2. Examples of elements ofD.A. Bouillard, J. Mairesse / C. R. Acad. Sci. Paris, Ser. I 339 (2004) 899–904 903ns.edFig. 3. The involutionψ . From left to right, the cases are 1,2,4, and 3.whereC is the set of cliques ofM(Σ̃, Ĩ ). Fig. 2 shows some alternate traces for the Basic Example.Theorem 4.1. In Z〈〈M(Σ̃ , Ĩ )〉〉, we have the following identity:( ∑t∈φ(F(Σ,I ))t)·( ∑d∈D(−1)|d |d)= 1. (7)Proof. We are going to use the same type ofbijective argument as in [17, Remark 5.2]. For a tracet ∈ M(Σ̃, Ĩ ),define Top(t) = {a ∈ alph(t) | ∃u, t = ua}. The set Top(t) contains all the pieces which are fully visible when theheapt is viewed from above. Givent ∈ M(Σ̃, Ĩ ) andh ∈ D, defineM(t,h) ⊂ Σ by M(t,h) = ι(Top(h) ∪ {a ∈Top(t) | ∀b ∈ Top(h), aĨb}). Observe thatM(t,h) = ∅ if and only if (t, h) = (1,1). We now define an applicatioψ :φ(F(Σ, I)) × D −→ φ(F(Σ, I)) × D.We haveψ(1,1) = (1,1). For (t, h) ∈ φ(F(Σ, I)) × D with (t, h) = (1,1), we distinguish between four caseEquipΣ with a total order and leta be the smallest letter ofM(t,h).1. If we havet = t1a1 with a1 ∈ {a, ā} anda, ā ∈ alph(h), thenψ(t,h) = (t1, a1h).2. If we haveh = a1h1 with a1 ∈ {a, ā} anda, ā ∈ Top(t), thenψ(t,h) = (ta1, h1).3. If we havet = t1a1 with a1 ∈ {a, ā} andh = a1h1, thenψ(t,h) = (ta1, h1).4. If we havet = t1a1 with a1 ∈ {a, ā} andh = ā1h1, thenψ(t,h) = (t1, a1h).The dynamics ofψ for the Basic Example is illustrated in Fig. 3. Here the order isa < b < c.Observe that the configurations of typeaa are allowed inφ(F(Σ, I)) but not inD, and the other way round thconfigurations of typeaā are allowed inD but not inφ(F(Σ, I)). Notice also that the smallest letters ofM(t,h)andMψ(t,h) are the same. It follows easily from these two points thatψ2 = Id. It is also clear that the only fixepoint ofψ is (1,1). Furthermore, if(t, h) = (1,1) andψ(t,h) = (t ′, h′), then we haveth = t ′h′ and|h′| = |h|± 1.So, we have(−1)|h|th = −(−1)|h′|t ′h′.SetS = φ(F(Σ, I)) × D. By decomposing the sum over the orbits ofψ , we get( ∑t∈φ(F(Σ,I ))t)·( ∑h∈D(−1)|h|h)= 1+∑(t,h)∈S\{(1,1)}(−1)|h|th = 1. Corollary 4.2. In Z[[x]], the following identity holds:GrF(Σ, I) · µM(Σ̃,Ĩ )(x/(1+ x)) = 1. (8)904 A. Bouillard, J. Mairesse / C. R. Acad. Sci. Paris, Ser. I 339 (2004) 899–904Theo-rvey, weisuentlyto thepect toctureouen,.4)mtive,Corollary 4.2 appears in Lewin [14], with a quite different proof from which the stronger statement inrem 4.1 cannot be deduced. So the growth series is a rational series with a simple combinatorial definition. ObsethatµM(Σ̃,Ĩ )(x) = µM(Σ,I )(2x). Since the growth rate is the inverse of the modulus of a dominant singularitobtain:ρF(Σ,I ) = ρM(Σ̃,Ĩ ) − 1 = 2ρM(Σ,I ) − 1. A consequence is that the results originally proved forρM(Σ,I ) in[10,12] can be directly transferred toρF(Σ,I ). For instance, GrF(Σ, I) has a unique dominant singularity whichpositive real. If(Σ, (Σ ×Σ)\I) is a connected graph, then this dominant singularity is of order 1 and conseq#{t ∈ F(Σ, I), |t| = n} ∼ cρnF(Σ,I ).We conclude by listing some complementary results, for details see [1,2]. The identity (7) can be liftedfree monoid under some conditions on(Σ, I) (this is a consequence of [4, Theorem 4.4.2]). Theheight h(t) of atracet is the height of the corresponding heap. More formally, the height is the length defined in (3) with resthe set of generatorsC \ {1}. Theaverage height λ = limn n−1(∑|t |=n h(t))/(∑|t |=n 1) is an explicitly computablealgebraic number. The proof uses Corollary 4.2 in a preeminent way.Example 2. Consider the Basic Example. Applying Theorem 4.1, the characteristic series ofφ(F(Σ, I)) is theformal inverse of 1− ∑x∈Σ̃ x(x̄x)∗(x̄ − 1) + ∑ι(x)=a,ι(y)=b x(x̄x)∗(x̄ − 1)y(ȳy)∗(ȳ − 1). Using (5), we haveGrM(Σ̃, Ĩ ) = (1 − 6x + 4x2)−1. Applying Corollary 4.2, we get the growth series GrF(Σ, I) = (1 + x)2/(1 −4x − x2). The growth rate of the trace group is(√5+ 2). The average height isλ = (25− 3√5)/20.References[1] A. Bouillard, Rapport de DEA : Le groupe de traces, LIAF research report 2002-13, Université Paris 7, 2002.[2] A. Bouillard, J. Mairesse, Generating series of the trace group, in: Z. Ésik, Z. Fülöp (Eds.), Developments in Language Theory, in: LeNotes in Comput. Sci., vol. 2710, Springer-Verlag, 2003, pp. 159–170.[3] P. Cartier, D. Foata, Problèmes combinatoires de commutation e réarrangements, Lecture Notes in Math., vol. 85, Springer, 1969.[4] V. Diekert, Combinatorics on traces, Lecture Notes in Comput. Sci., vol. 454, Springer-Verlag, 1990.[5] V. Diekert, G. Rozenberg (Eds.), The Book of Traces, World Scientific, 1995.[6] C. Droms, B. Servatius, H. Servatius, Groups assembled from free and direct products, Discrete Math. 109 (1992) 69–75.[7] C. Duboc, Commutations dans les monoïdeslibres : un cadre théorique pour l’étude du parallélisme, PhD thesis, Université de R1986. Also LITP Report 86-25, Université Paris 7.[8] G. Duchamp, D. Krob, Partially commutative Magnustransformations, Int. J. Algebra Comput. 3 (1993) 15–41.[9] D. Epstein, J. Cannon, D. Holt, S. Levy, M. Paterson, W. Thurston, Word Processing in Groups, Jones and Bartlett, Boston, 1992.[10] M. Goldwurm, M. Santini, Clique polynomials have a unique root of smallest modulus, Inform. Process. Lett. 75 (3) (2000) 127–132[11] G. Hardy, E. Wright, An Introduction to the Theoryof Numbers, fifth ed., Clarendon Press, Oxford, 1979.[12] D. Krob, J. Mairesse, I. Michos, Computing the average parallelism in trace monoids, Discrete Math. 273 (2003) 131–162.[13] G. Lallement, Semigroups and Combinatorial Applications, Wiley, New York, 1979.[14] J. Lewin, The growth function of a graph group, Commun. Algebra 17 (5) (1989) 1187–1191.[15] G.-C. Rota, On the foundations of combinatorial theory. I. Theory of Möbius functions, Z. Wahrscheinlichkeitstheor. Verw. Geb. 2 (196340–368.[16] A. Vershik, S. Nechaev, R. Bikbov, Statistical properties of locally free groups with applications to braid groups and growth of randoheaps, Commun. Math. Phys. 212 (2) (2000) 469–501.[17] G.X. Viennot, Heaps of pieces, I: Basic definitions and combinatoriallemmas, in: G. Labelle, P. Leroux (Eds.), Combinatoire Énumérain: Lecture Notes in Math., vol. 1234, Springer, 1986, pp. 321–350.

Möbius inversion formula for the trace group

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