Algebraic and topological theory of languages

RAIRO - Theoretical Informatics and Applications, Jan 1995

J. Rhodes, P. Weil

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Algebraic and topological theory of languages

Informatique théorique et applications, tome 29, no 1 (1995), p. 1-44. <http://www.numdam.org/item?id=ITA_1995__29_1_1_0> L'accès aux archives de la revue « Informatique théorique et applications » implique l'accord avec les conditions générales d'utilisation (http://www.numdam. org/legal.php). Toute utilisation commerciale ou impression systématique est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. - Algebraic and topological theory of languages Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ (vol. 29, n° 1, 1995, p. 1 à 44) ALGEBRAIC AND TOPOLOGICAL THEORY OF LANGUAGES (*) by J. RHODES C1) and P. WEIL (2) Communicated by J.-E.PIN Résumé. — On dit qu'un langage est de torsion (resp. de torsion bornée, apériodique, apériodique borné) si son monoïde syntaxique est de torsion (resp. de torsion bornée, apériodique, apériodique borné). Nous généralisons les théorèmes sur les langages rationnels de Kleene, Schützenberger et Straubing pour décrire les classes des langages de torsion, de torsion bornée, apériodiques et apériodiques bornés. Ces descriptions imposent la considération de limites de suites de langages et d'automates pour certaines topologies définies par des filtrations du monoïde libre. Nous donnons également un théorème concernant les langages arbitraires sur des alphabets finis. INTRODUCTION The aim of this paper is to generalize the central results of the theory of rational, or recognizable languages (the languages which are recognized by finite automata) to a much wider class of languages over finite alphabets. We rely in part on the powerful algebraic methods whose use is wellestalished for recognizable languages. In that more restrained framework, the relevant algebraic objects are the finite monoids. A standard algebraic way of generalizing finiteness is the concept of torsion: an algebraic object Both authors gratefully acknowledge support from the first author's National Science Foundation grant DMS88-03362. The second author was also supported in part by the Projet de Recherche Coordonnée "Mathématiques et Informatique". (*) Received February 1987, revised February 1993. C) Departement of Mathematics, University of California, Berkeley, CA 94720, USA. E-mail: (2) L.I.T.P., Institut Biaise Pascal, C.N.R.S., 4, place Jussieu, 75252 Paris Cedex 05, France. is torsion if each of its éléments has only finitely many distinct powers, and it is bounded-torsion if the number of distinct powers of its éléments is uniformly bounded above by some fixed integer. Around 1900 Burnside conjectured that ail finitely generated bounded-torsion groups were finite. This was proved false 70 years later by Adjan-Novikov when the exponent is large and odd. More recently a shorter proof was given by Ol'shanskiï using small-cancellation diagrams (see [12]). More recently also, some important results on bounded-torsion monoids were obtained by Mc-Cammond [11], de Luca and Varricchio [2, 3] and Pereira do Lago [14]. Our main theorems deal with bounded torsion and torsion languages, that is, languages which are recognized by bounded torsion or torsion monoids. Such monoids were already considered by Rhodes [17, 18]. Another essential tool of our work reveals interesting connections with topology. Let X be a finite alphabet. We say that a séquence 1 = (In)n of subsets of the free monoid X* is a filtration if IQ- = X*, In+i Q In for ail n > 0 and C\nIn — 0. Now a filtration X gives lise to a topology on X* U{oo} for which a basis of open sets is {{w}|w G X*} U {In U {oo}|n > 0}. In particular a séquence of words (wn)n tends to oo if and only if, for each n > 0, ail but a finite number of the w^ lie in In. This topology can then be extended to define the convergence of séquences of languages in X* and of séquences of automata over X: a séquence of languages (Ln)n tends to a language L modulo I if for each n > 0, there exist k > 0 such that Lm\In = L\In for ail m > k. In an analogous fashion, a séquence of automata (An)n tends to an automaton A modulo 1 if for each n > 0, there exists k > 0 such that, whenever m > k and u, v £ In, (<^ . u — q™* v) <£• (qo * u = qo - v) (where q™ is the initial state of Am and ço the initial state of A}. Note that this notion of convergence modulo a filtration arises in an intuitive fashion when one considers classical machine models such as, say, Turing machines. For this model, we can consider bounding the amount of time or space or any suitable function of time and space, which we can call "stuff', made available to the machine. Any Turing machine M, when restricted to using at most n units of stuff, is equivalent to a finite-state machine Mn, and it is natural to try and view the language recognized by M as "a limit" of the (rational) languages recognized by the Mn. If we let, for ri > 0, In be the set of ail words w G X* such that M cannot make any décision using less than n units of stuff upon reading w, then I — (In)n is a filtration and the language of M is the limit of the languages of the Mn with Informatique théorique et Appiications/Theoretical Informaties and Applications respect to J . This idea is only intuitively presented hère, but it is illustrated by a surprising result proved in the second appendix of this paper. Our main results are characterizations of the classes £tor, £btor> £ap and £bap> respectively of all torsion, bounded-torsion, aperiodic and boundedaperiodic languages. (A monoid S is aperiodic if for each s E S there exists n > 1 such that sn — sn+1 and a language is aperiodic if it is recognized by an aperiodic monoid.) These characterizations generalize the theorems fo Kleene, Schützenberger and Straubing on recognizable languages. Recall that these theorems state that the classes of rational (resp. rational aperiodic, rational subgroup-solvable) languages are the least classes containing the finite languages and elosed under certain language opérations (such as union, product, star, etc.) Our results characterize the classes Aor* ^btor» ^ap and Aap as the least classes containing the finite languages, elosed under some of these languages opérations, and elosed under taking certain limits with respect to certain filtrations. Of course these results hint at a possible generalization of Eilenberg's variety (or stream) theorem to classes of arbitrary languages (not just rational), and we explore and prove this generalization in our third appendix. Part of the proof of these main results relies on combinatorial and algebraic methods more or less of the same flavor as the techniques used classically to deal with rational languages. The second part of these proofs uses in a crucial way the properties of the finitely generated Burnside monoids, recently established by McCammond [11] and Pereira do Lago [14], and the properties of two semigroup expansions which were studied by Birget, Rhodes and Henckell [1, 17, 6]. One of these expansions, S H^ S^ is particularly interesting to illustrate another point of view on the generalization of finiteness, a concept underlying all of this paper. Given any finitely generated monoid 5, a finitely generated monoid S^ can be constructed along with an onto morphism ?r from S^ onto S such tiiat: (1) S^ is "close" to S (technically, the inverse image by n of each idempotent of S satisfies the identity xb — #6), and (2) S^3) contains a séquence of ideals (Jn)n such that JQ = S^\ Ç Jn for all n and nn Jn — 0, and such that each Rees quotient is finite. This séquence of ideals (Jn)n is of course reminiseent of our définition of a filtration over a free monoid. Then, if we consïder the topology on Ê^ U { oo} defined as above ^with {{s}\s e P } U {Ju U-{oo}|n > 0} as a basis of open sets), then S&) U {oo} is cmnpacL Note that compactness is another voL 29, n° I, 1995 natural generalization of finiteness. In fact, the resuit which we present in our second appendix is essentially an application of this construction, and we give its full details in that appendix. The précise organization of this paper is the following. In Section 1, we present rapidly the basic définitions and properties of automata, semigroups and syntactic monoids, and review the statement of the theorems of Kleene, Schützenberger and Straubing. Section 2 is devoted to exploring the first properties of torsion and aperiodic monoids and languages. In Section 3 we introducé the notions of convergence of a séquence of languages and of a séquence of accessible automata modulo a given filtration. Our main theorems are stated in Section 4 and proved in Section 5. We then consider in three subséquent appendices some connected results. The first one gives a variant of our results in terms of convergent séquences of onto morphisms. The second one is the description of a rational filtration which can be canonically associated to an arbitrary congruence on X*, and the last one is the generalization to arbitrary classes of languages (over finite alphabets) of Eilenberg's variety theorem. We wish to acknowledge the special debt owed to Douglas Albert whose knowledge of computer science and insights he has so generously shared. Also this paper sterns from a preliminary reprint of the same title as this paper, published by the first author as a report by the Center of Pure and Applied Mathematics of the University of California as MAP-180 in September 1983. 1. PRELIMINARIES In this section we will review briefly the définition and basic properties of the objects that we will be dealing with, namely languages and automata. In particular we will remind the reader of the concept of recognizability of a language by a monoid, and we will recall some of the fondamental results of the theory of rational languages. For a more detailed présentation of the various aspects of the theory of languages and automata, the reader is referred to [13, 4, 10, 16]. 1.1. Languages and automata Throughout this paper, X will dénote a finite non-empty set called the alphabet. lts element are called letters. Finite (possibly empty) séquences of letters are called words. The set X* of all words over the alphabet X is a monoid under concaténation. lts identity is the empty word, denoted 1. The monoid X* is the free monoid over X. A language over X is any subset of X*. An automaton over X is a 5-tuple A — (Q, X, qo, A, F ) where Q is a countable (no/ necessarily finité) set called the state set, qo G Q is called the inirifl/ state, À : Q x X —> Q is the transitionfunction and F Ç Q is called the set of final states. When there is no ambiguity as to which automaton is being discussed, we write À (ç, x) = q •x (q G Q and x G X). The function À is extended to A : Q x X* —> Q by letting q * 1 = q for q . wo; =z (g . w) • x all q G Q, for all g G Q, x G X and tu G X*. We will always suppose our automata to be accessible, that is, they satisfy qo *X* = Q. The language recognized by A is L(A) = {w G X* |ço • w € F } . Let L Ç X* be a language. The translates of L are the languages vrlL Lu'1 u^Lv-1 = {ve x*\uv eL} (ue x * ) , = {ve X*\vu eL} = (u~1L)v-1 ^u"1 (ue X * )a n d ( L Î T 1 ) (u, v G X*). It is easy to verify that, for each u, v G X*, then n " 1 (v~1L) — (vu)~1L. Let us note also the following simple remark. If ^o = (Q, X, qo, A, F) is an automaton, if u G X* and q\ — qo • n, let Ai = (Q, X, ci, A, F ) . Then L (Ai) = u~lL (Ao). To each language L Ç X* we associate a canonical automaton A (L) = (Q, X, qo, A, F) in the following way: Q = {u^Llu G X*}, qo= L = l - a L , F = {tT^Llu G X* and 1 G u^L}, A(u~1L, x) = (ux)~lL for u G X* and x G X. In this automaton, for ail w G X*, we have q0 . w — (1~1L) • w — w"lL. Therefore w e L (A (L)) if and only if 1 G w~1L, that is, if and only if w e L. Thus L (A (L)) — L. This shows in particular that every language is recognized by some automaton. In gênerai, a language can be recognized by several different automata. However, A (L) is the minimal automaton of L in the following sensé. PROPOSITION 1.1: Let L Ç X*, let A{L) — (<QL, X, L, \ L , FL) and let A — (Q, X, go> A, F ) i?e any automaton recognizing L. Then there exists a surjective mapping TC : Q —> QL such thaï qotr = L, FIT — FL and qir - u — (q • u) TT for ail q E Q and u E X*. (We say thaï A reduces to A (L).) Furthermore, if B is another automaton such that each automaton recognizing L reduces to B, then B is isomorphic to A{L). Proof: Well-known. D In the sequel it will be convenient to consider automata with unspecified set of final states, that is, of the form {Q, X, go> A). We still call these objects automata and we let A (X) be Üie class of all automata with unspecified set of final states over X. If A E A (X), we let C (A) be the set of all the languages that are recognized by A (when the set of final states assumes ail possible values). 1.2. Syntactic monoids We first recall a few basic notions on semigroups. Readers are referred to [4, 10, 16, 8] for more details. A semigroup is a pair (5, •) where S is a set and * is an associative binary opération on 5. In a semigroup 5, an idempotent is an element e such that e2 —e. An identity (resp. zero) is an idempotent e such that es — se = s (resp. es = se = e) for ail s E S. An identity (resp, zero) is usually denoted 1 (resp. 0). Any semigroup has at most one identity (resp. zero), but it may have an unrestricted number of idempotents. We say that 5 is a monoid if it has an identity. If S is a semigroup we define S1 to be the monoid equal to 5 if S is a monoid, and to S\J {1} otherwise (where 1 is an adjoined identity). If A, B Ç S, we let AB - {ab\a E A, b E B}, Then the power set of S, V (S), is a semîgroup. If T Ç S and T2 Ç T, we say that T is a subsemigroup of 5. Let A Ç S. The subsemigroup (A) generated by A is the least subsemigroup of S containmg A, that is, the set \J An of all finite n>\ products of éléments of A. We say that A générâtes S if S = (A) and that S is finitely generated if S is generated by a finite set. An idéal of a semigroup S is a subset I of S such that S1!S1 —/ . If / is a non-empty idéal of 5 , a new semigroup ( 5 / J , *) is constructed, called the Rees quotient of S by 7, by letting S/I = S\I U {0} and, for 5, s1 E S/I, O otherwise. ALGEBRAIC AND TOPOLOGICAL THEORY OF LANGUAGES. If S and T are semigroups, a morphism y? : S -»• T is a mapping such that (ss') (p— (s(p) (s'cp) for all s% s1 G S. When S and T are both monoids, we will implicitly assume that l(p — 1. We say that T divides S if there exists a surjective morphism from a subsemigroup of S onto T. If / is a non-empty ideal of 5, then the canonical projection TT : S —> S/I defined by f 0 otnerwise. is an onto morphism. A congruence on a semigroup S is an équivalence relation ~ such that, for all s, s' G 5 and u, v G S 1 , 5 ~ s' implies usv ~ US'Ï;. If <p : 5 —> T is a morphism and if we define ~ ^ on 5' by s ~ ipSfif and only ifs<p — ^V» then ~ ^ is a congruence. Conversely, if ^ is a congruence, then the set S f ^ of ^-classes is naturally equipped with a semigroup structure given by [s]* [sf] — [ss'J for all s, sf G 5 (where [s] is the ~-class of s). The canonical projection from 5 onto 5 / ~ is a morphism. Morphisms on free monoids will be crucial in the sequel. Let S be a monoid and let X be an alphabet. Recall that, for any mapping <p : X —» S, there exists a unique morphism from X* into 5 extending (p. Let A — (Q, X, go, A) G v4(X) be an automaton. For each word u G X*, let u/x be the function from Q to Q given by g i->- g * w. Then fi defines a morphism from X* into the monoid of fonctions from Q into itself ( a monoid under the compositions of fonctions). We say that n is the transition morphism of A and we dénote the mage of \i by S (A), the transition monoid of A Note that JJL and 5 ( A) depend only on Q, X and À. Let L be a language recognized by A with set of final states F, and let P be the set of fonctions from Q to Q which map ^ to an element of F . Then L — PpT1. In gênerai we say that a morphism ip : X* —» 5 recognizes a language L C X* if there exists a subset P of S such that L = Pip"1. Thus, if an automaton recognizes L, then its transition morphism recognizes L as well. Conversely, let cp : X* —* 5 be an onto morphism recognizing a language L. Then £ — P<p~x for some P Ç 5. We define an automaton A (S) = (S1, X, 1, A) G A (X) by letting A (s, x) = 5 (ar^) (5 G S 1 and a: G X). If we choose P for the set of final states, then this automaton recognizes L. Let L Ç X*. The syntactic congruence = £, in X* is the largest (coarsest) congruence for which X is a union of classes. One can easily prove (see for instance [4, 10]) that ~ L in is given by w\ = 1W2 if and only if Vu, v G X*, uwi v G L <=ï UW2 v £ L. The quotient monoid X * / = L is called the syntactic monoid of L and is denoted S (L). The syntactic morphism of L is the canonical projection from X* onto S (L). By définition of = £,, the language L is recognized by its syntactic morphism. Furthermore, the following results are well-known (see [4, 16]). SLT means S «-< T. PROPOSITION 1.2: Ler L Ç X*, and ter T]L : X* —>5 (L) &e /te syntactic morphism. (1) r}L is the transition morphism ofthe minimal automaton A (L) of L. (2) If a morphism cp : X* —»•5 recognizes L, r/zen 5 (L) < S. Let L i , . . . , L n be finitely many languages in X*. We define BT (L\,..., Ln) to be the least family of languages of X* containing the languages of the form i t " 1 ^ ^ " 1 (1 < i < n and u9 v G X*) and closed under complement and arbitrary unions and intersections. PROPOSITION L3: (Little boxes theorem) Let L i , . . . , Ln be finitely many languages in X* and let r}i : X* —>S (Li) be their respective syntactic morphisms. Let 77 be the morphism from X* into S (L\) x . . . x S (Ln) defined by wq = (wrji,..., wqn). Then L E BT (L\,..., Ln) if and only if L is recognized by 77. Proof: The proof relies on the following important remark: Let w E X*. Then where the first union runs over all pairs of words (u, v) such that uwv G Lj, and the other union runs over all pairs (#, y) such that xwy £ Li. This is immédiate by the characterization of the syntactic congruence. n Now, if lu G X*, then wr}r]~l — f^wrarj'1. So the languages recognized t=i by 7] are ail in BT{L\,..., Ln). Conversely let us remark that if wrji = wfrn and w G u~1LiV~1 for some i, u and v, then vJ G u~1LiV~l. That is, each u~1L{V~1 is recognized by the morphism ru and hence by the morphism rj. On the other hand, any Boolean combination (using complement and arbitrary unions and intersections) of languages recognized by 77 is also recognized by 77. So every language in BT ( L i , . . . , Ln) is recognized by 77. D COROLLARY 1.4: 77^ ser o/ languages recognized by the minimal automaton of a language L is BT (L). 1.3. Rational languages If L, L' Ç X*, we define the product LL1 and the star L* by LLf — {wwf\w G L, wf E L} and L* = {1} U tüi .. .wn\n > 1, u^ E Lfor all z}. The class of rational languages over X is the least class of languages containing the finite subsets of X* and closed under union, product and star. The following theorem, due to Kleene [9] is a fundamental result of the theory of rational languages. THEOREM 1.5: (Kleene's theorem) Let L Ç X*. The following are equivalent. (1) L is rational. (2) L is recognized by a finite automaton. (3) L is recognized by a finite monoid. ( 4 ) A (L) is finite. (5) S (L) is finite. Proposition 1.3 implies immediately the following corollary. COROLLARY 1.6: /ƒ L i , . . . , Ln Ç X* are rational languages, then BT{L\,..., Ln) is a finite set of languages, all ofwhich are rational. An important subclass of the rational languages is the class of star-free languages: it is the least class containing the finite sets and closed under Boolean opérations and product. Schützenberger [19] gave the following characterization of the star-free languages. We say that a monoid S is aperiodic if VseS, 3 n > l , s n + 1 = s n . THEOREM 1.7: (Schützenberger's theorem) Let L Ç I * . The following are equivalent. (1) L is star-free. (2) L is recognized by a finite aperiodic monoid. (3) S (L) is finite and aperiodic. Again we have COROLLARY 1.8: If L\,.,., Ln Ç X* are star-free languages, then all the languages in BT {L\.,..., Ln) are star-free. Because of Schützenberger's theorem, star-free languages are also called rational aperiodic languages. It is not difficult to verify that a finite monoid is aperiodic if and only if its subgroups are trivial. Two other subclasses of the rational laguages will be of interest for us. We say that a language X is rational subgroup-cyclic (resp. rational subgroupsolvable) if each subgroup of its syntactic monoid is cyclic (resp. solvable). In particular each star-free language is rational subgroup-cyclic and each rational subgroup-cyclic language is rational subgroup-solvable. If L Ç X*, if p is prime and 0 < q < p, we define (L, p, q) to be the set of ail words w having a number of préfixes in L congruent to q mod p. Straubing [20] proved the following characterization of rational subgroup-solvable languages. THEOREM 1.9: (Straubing's theorem) Let L Ç X*. The following are equivalent. (1) X can be obtained from the finite subsets of X* using only Boolean opérations, products and opérations of the form L i-> (X, p, q) where p is prime and 0 < q < p. (2) X is recognized by a finite monoid ail ofwhose subgroups are solvable. (3) X is finite subgroup-solvable. COROLLARY 1.10: IfLi,...,LnÇ X*are rational subgroup-solvable, then all the languages in BT ( X i , . . . , Ln) are rational subgroup-solvable. 2. TORSION AND APERIODIC LANGUAGES AND SEMIGROUPS Let S be a semigroup. We say that S is torsion if Vs G S, We say that S is n-bounded-aperiodic for n > 1 if sn — sn+l for all s e S. Finally we say that S is bounded-aperiodic if 5 is n-bounded-aperiodic for some n. We dénote by Ap, bAp and bAp„ the classes of aperiodic, boundedaperiodic and n-bounded-aperiodic semigroups. Notice that finite aperiodic semigroups are bounded-aperiodic. The following important properties are easily verified. PROPOSITION 2.1: The following strict containments hold: hApn C b A p n + i , b A p n C bTorn, bTor^ C bTor^+i, bTor^ D A p — b A p n bAp C bTor, A p C Tor. If T divides S and S is in Tor (resp. Ap, bTor, bAp, bTor„, bAp„), then so is T. The classes Tor, bTor, Ap and bAp are closed under finite direct product and the classes bTor„ and bApn (n > 1) are closed under arbitrary direct product We will also use the following property of the classes Tor, Ap, bTor, bAp, bTorrt and bApn. Let ?r : S —* T be a morphism. We say that ir is torsion (resp. aperiodic, n-bounded-torsion, n-bounded-aperiodic) if, for each idempotent e of T, the semigroup eTr"1 is in Tor (resp, Ab, bTorrt) bAprt). We say that n is bounded-torsion (resp. bounded-aperiodic) if there exists n > 1 such that TT is n-bounded-torsion (resp. n-bounded-aperiodic). Note that this is different from requiring that evr"1 G bTor (resp. bAp) for each idempotent e of T. PROPOSITION 2.2: Let -K : S —> T be a morphism. (1) If T G Tor (resp. Ap) and TT is a torsion (resp. aperiod) morphism, then S G Tor (resp. Ap). (2) Let n, n' > 1. If T G bTorn (resp. bApn) and ir is a n*-boundedtorsion (resp. nf -bounded-aperiodic) morphism, then S G b T o r ^ ' (resp. (3) If'T E bTor (resp. bAp) and ir is a bounded-torsion (resp. boundedaperiodic) morphism, then S G bTor (resp. bAp). Proof: We prove (1) for torsion semigroups and morphisms. The other proofs are similar. Let ir : S —*T be a torsion morphism with T G Tor. Let s G S. Then (sir)n ~ (S7r)2ri for some n > 1. Thus e = snir is an idempotent of T and sn G eTr"1. Since eTr"1 G Tor, there exists n' > 1 such that (sn)n' = (sn)2n'\ that is snn' = 52ri7l/. So 5 G Tor. . D We now turn to languages. We say that a language L Ç X* (resp. aperiodic, bounded-torsion, bounded-aperiodic, n-bounded-torsion, nbounded-aperiodic (n > 1)) if it is recognized by some automaton A such that (S) A G Tor (resp. Ap, bTor, bAp, bTorn, bAp„). We dénote these classes of languages by £ t o r , £ a p , A>tor, Aap, ^btor' £baP- O f c o u r s e we have - M>tor > ^btor ' ' ^ a P ~~ ^bap r- r, r r- r ap ^- -Motor > ^ a p ^- *^tor* Since rational languages are recognized by finite automata, ail rational languages are bounded-torsion and ail star-free languages are boundedaperiodic. Note also the following proposition, which is immédiate using Propositions 1.2 and 2.1. PROPOSITION 2.3: Let L Ç X*. L is torsion (resp. bounded-torsion, n bounded-torsion, aperiodic, bounded-aperiodicy n-bounded-aperiodic) if and only if it is recognized by a monoid in Tor (resp, bTor, bTorn, Ap, bAp, bAprt), if and only if S (L) is in Tor (resp. bTor, bTorrt, Ap, bAp, bApw). It is well-known that not ail torsion languages are aperiodic. For instance the language of all words of even length on an alphabet X is rational but not aperiodic. The set of ail square-free words over a 3-letter alphabet is an aperiodic language which is not rational. More precisely, this language is in A p . This is a conséquence of the existence of an infinité square-free word over a 3-letter alphabet. Torsion and bounded torsion groups have received considérable attention. Examples of finitely generated torsion groups which are not bounded-torsion were exhibited by Golod and Shafarevitch (see Herstein [7]), by Grigorchuk [5], etc. These provide examples of torsion languages which are not boundedtorsion. The existence of infinité (Burnside) groups with two generators and fixed exponent k (for k large enough and odd) provides other examples of languages that are bounded-torsion and not rational. See [12]. Finally, not ail languages are torsion. For instance the Dyck language D over two letters, defined by Do = {1}, Dn+i = {bD*na)* and D = | J Dn is not a torsion language. 1 3 Let w be a non-empty word. Recall that w* = {wn\n > 0} and that w is primitive if w E u* implies w = u. In particular each non-empty word is a power of a primitive word. For L Ç X* we define K (L, w) by ÜC (L, w) — {k > 0\wk E L}. Let n > 1. We say that a set X of integers is ultimately n-periodic if there exists t > 0 such that A; > £ and k E K implies k + n € K. We say that K is ultimately periodic if it is ultimately n-periodic for some n. When n ^ l w e speak of an ultimately aperiodic set of integers. The least t for which the above implication holds is called the threshold. We have the following characterization of the classes Aor, Atp, ^tor and £apJ. PROPOSITION 2.4: Lef L Ç X* and let n > 1. (1) L G Aor (ƒ and on/y # /or a/Z L' G £ T (L) and for ail w E X*, L7 fi u>* z*s rational and aperiodic, which is equivalent to requesting that L' n tü* E Aor or that K (L', w) be ultimately periodic. (2) L E £ a p *ƒ and only if, for all Ü E BT (L) a n d / o r a// w E X*, L' fi w* /s rational and aperiodic, which is equivalent to requesting that Lf Hw* E £ap or that K (L', tu) èe ultimately aperiodic. (3) L E 4ôr '/ a t ó o n ^ ^ /^r a / / L / G 5 T (L) a w d / ö r aW ^ E X*, K ( I / , tu) z's ultimately n-periodic with threshold n. ( 4 ) L E £ap *ƒ and only if for ail L1 E BT (L) a n d / o r a/Z w E X \ ÜT (L', tu) w ultimately aperiodic with threshold n. (5) Statements (l)-( 4 ) above still hold ifwe restrict the words w to being primitive. Proof: Let us prove (1) and the corresponding assertion of (5). Let us first assume that L E Aor, and let L' E BT (L) and w e X*.By Corollary 1.4, L1 is recognized by the syntactic morphism of L and by Proposition 2.3 this implies that V E Aor- Therefore there exists n > 1 such that wn = w2n where = is the syntactic congruence of L', and hence wk = wk+n for ail k > n. Thus K (L', w) is ultimately n-periodic with threshold n. Therefore Lfnw*^{wk\k<n&ndwk eLf}U ( J n<fc<2n «;*(«;")* is rational (and hence torsion). For the converse let us assume that L £ Aor and let 17^ be its syntactic morphism. Then there exists w E X* such that the wnr}L are pairwise distinct. We can further assume that w is primitive. Let now K be a nonultimately periodic set of integers, say K is the set of all primes, and let p - = {wkr]L\k G K}. Let 1/ = Pril1. T h e n L' i s recognized by T/L, SO that L' G BT (L). But K (Lf, w) = K which is not ultimately periodic. The first part of this proof can be easily adapted to show that if L G £ a p (resp. ^bitor' ^bap)' t^ien ^ ( ^ w) *s ultimately aperiodic (resp. ultimately n-periodic with threshold n, ultimately aperiodic with threshold n) for all II G BT(L) and W G P , that is, the direct part of statements (2) to ( 4 ). For the converse part in statement (2), we consider L £ £ a p and TJL its syntactic morphism. We already know that if L £ £ t o r then we can find V G BT (L) and w a primitive word such that K (L', w) is not ultimately periodic. Let us now assume that L G £tor\Aip- Then there exists a word w in X* (which can be assumed to be primitive) such that w*r}i consists of exactly a + b éléments 1, WT)^, . . . , wa7?i,, ^a+1ï?x/,...... w01^"1^ with i y a % = wa+br)L, a > 0 and 6 > 2. Let P = zt;ar7L and 1/ = Frç^1. Then Ll recognized by r)L and hence H G B T (£). Furthermore K (Lf\ w) = {a + &^|A; > 0}, which is not ultimately aperiodic. Similarly, for the converse part of statement (3) (resp. ( 4 )), it is sufficient to consider L G Aor\^t>tor (resP- ^v\^hL> an{^ ^L *ts s y n t a c t i c morphism. Then there exists w G X* (which can be assumed to be primitive) such that wnr}L ^ w2nT]L (resp. wnr]L ^ ^ n + 1 % ) . If we let P = wnr}i and = Pri^1, t n e n L* E BT (L) a n d K (L'i w) c o n t a i n s n b u t n o t 2 n (resPV n + 1), so K (Z/, w;) is not ultimately n-periodic (resp. ultimately aperiodic) with threshold n. Q Finally we note the following characterization of automata whose transition monoid is torsion or aperiodic. PROPOSITION 2.5: Let A = (Q, X, g0, A) € ^4 (X). r/zen £ (.4) Ç £ t o r £ap) if and only if S (A) G Tor (resp, Ap). If S (A) G T o r (resp;. Ap), then C (A) Ç £ t o r (resp. £ a p ) by définition. We prove the converse statement concerning torsion languages. This proof can easily be modified to prove the aperiodic case. Let fj, : X* —• S (A) be the transition morphism of A We assume that S (A) g Tor and we wil! prove that £ (A) <£ £tOr- Since S (A) g Tor, there exists w G X* such that the wnjjb are pairwise distinct. First case. a b 3q G Q, Va, b > 1, q • vf £ q • w + . Let F = {q •• iüp|prime} and let £ be recognized by A with F as the set of final states. Let u G X* be such that qo - u — q and let L' — u~1L. Then 1 5 K (L', w) = {k > l\uwk G L} is the set of prime numbers, which is not ultimately periodic. So Proposition 2.4 shows that L is not torsion. Second case. \/q G Q, 3a, b > 1, q • wa = q - wa+b. For each q G Q let C9 be the "cyclic" part of q-w*9 that is, Cq = {g' G Q\3a, b > 1, q •wa = g •wa+6 = <?'}. Let a^ > 0 be minimum such that q * waq E C^ and let bq > 1 be minimum such that q-waq — q-waq^bq. Note that (7^ = {g-iün[n > a^} has cardinality bq. Furthermore the sets Cq are pairwise disjoint. Since 5 (A) g Tor, either the set {aq\q G Q} or the set {bq\q G Q} is unbounded. Therefore we can choose a séquence (qn)n of states such that we have either : or : aqi < aq2 < ... < aQn < .... bqi<bq2< ...< bQn < ... Let us first assume that aqi < aq2 < ... Let L be the language recognized by A with set of final states I ) Cqn and let = L be its syntactic congruence. n>\ Note that if u G X* and qo *u = qn, then uwm E L ïf and only if m> aQn. If L is torsion there exists a, b > 1 such that wa = ^XÜ0"1"6., Let us choose n and m such that a < aqn < a + rn&, and let u G X* be such that qo-u — qn. Then uwa 0 £ and uwa+mb G L, thus contradicting tüa ~ LWa+b. So L is not torsion. Let us first assume that bqi <bq2 < ... Let L be the language recognized by A with set of final states {qn • wa^ \n > 1} and let = ^ b e its syntactic congruence. We notice that if u G X* and qo - u = qn •wa<ln then utüm G L if and only if m = 0 (mod 6?w). As before, if L is torsion there exists a, &> 1 such that wa = jjWa+b. Let n be such that a + 6 < 6 ^ and let c = b9w - (a + 6). Then a + & + c = 0 (mod 6^) but a + 26 + c ^ 0 (mod &9n). Therefore if go • « = qn * ^Ö9n ( w e l * ) then uwa+b+c G L and contradicting wa ~ iwa+b. So L is not torsion. • 3. FILTRATIONS AND LEMITS OF SEQUENCES OF LANGUAGES AND MORPfflSMS This section is devoted to the concept of a topology on X* defined by a filtration and to the notions of convergence of séquences of languages and of séquences of automata over X. 3.1. Topology induced by a filtration Let X be a finite alphabet. A filtration X of X* is a séquence of languages X = (In)n>o such that and such that ( j In — 0. We say that X is in a class of languages C if each 71>0 In is in £ . We will be interested in particular in rational filtrations (each In is a rational language) and in idéal filtrations (In = X*InX* for each n > 0). Let X* be the monoid X* = X* U {oo} consisting of X* and a new element oo such that x (oo) — (oo) x — oo, for all x G X*. Note that oo is a zero of X* and that X* is a submonoid of X*. It is classical to consider the topology Topj on X* defined by a filtration X — (In)n on X*. For this topology, a basis for the open sets is {{w}\w G X*} U {/„ U {oo}|n > 0}. For example, let N = {1, 2 , . . . , n , . . . } U {oo} be the one-point compactification of the positive integers: a basis for the open sets of N is where Fn {{n}\n > 1} U {Fn U {oo}|n > 1} — {i G N|z > n } . If X = (In)n is a filtration on X*, we define a fonction r% : X* —• N by TX (oo) = oo and, for w G X*, rj (w) = min {n > 0\w $ In}. In other words, if w G X*, then TX (w) — n w G Ijç for ail k < n, and w £ 1^ for all k > n. Therefore r ^ 1 (jPnU{oo}) = /nU{oo}. In particular, Topj is the last topology on X* that makes r j continuous. It is easy to verify that a séquence (wn)n of X* converges to w if and only if it is eventually stationary at w, or w = oo and lim rx (w) = oo n (Le., the séquence (wn)n "falls further and further down in X"). If X is an idéal filtration, then one can verify that X* is a topological monoid (the multiplications in X* is a continuous fonction). Furthermore, for each k > 0, X*/{I^ U {oo}) is a monoid, and if this monoid is endowed with the discrete topology, then the projection TT^ : X* —> X* /(Ik U {oo}) is a continuous morphism. For any filtration I on X*, one can defined a ultrametric distance function dj on X* letting dj (tü, w) = 0 for all w G X*, dj (tv. oo) = —r for all w G X* rr(jw(w) ) dj (w\, tü2) = max(dx (IUI, oo), for all distinct w\, and , OO)) in X*. Note that, for all n > 1 and ^ i 7^ w% in X*, we have y W2) < l/n G /nFor w G X* and e > 0, let B (w, e) be the open bail with center w and radius e. Then we have B (oo, e) = In U {00} where n = min < k — < e> and for w G X* {w} 1 K e) = (00, e) = In U {00} where n = min < k — < e> if e < i f e > Therefore, the topology induced by dj on X* is exactly Topj. 3.2. Séquences of languages We will be interestêd in languages of X* which are limits in a certain sense of a séquence 'of languages with respect to a given filtration X on X*. More precisely, the metric dj induces a Hausdorff metric on the set of closed subsets of X*: if L and L' are closed subsets of X*, then Dj (L, Lf) is defined by Dj (L, l!) — minje > 0 | \/w G L, 3w' G L', d j (u;, u/) < e and Viu' G L7, Biu G L, dj (w, w1) < e}. Note that a set L Ç X* is closed if and only if oo G L or L Ç X*\In for some n > 0. In particular, if L Ç X*, then L U {00} is always closed, and the closure of L is either L or L U {00}. If ( L n ) n is a séquence of languages of X* and I Ç X*, we will write L = l i m i n if L U {00} is the limit of the séquence of closed spaces (Ln U {oo})n, that is, if lim Dj (Ln U {00}, L U {00}) = 0. One can verify the following proposition. PROPOSITION 3.1: Let L, Ln Ç X* and let X = {In)n be a filtration on X*. Then L — liniLn Vn, 3k > 0, Vm > &, Lm\In — L\In. Exarnple: Let L Ç X* and let In (n > 0) be the set of all words of length greater than or equal to n. Let Ln ~ L\In for each n > 0, and let X — ( / n ) n . Then X is a cofinite ideal filtration, each Ln is finite and L = lim xLn. Note the following easy property of thîs topology on the set of languages. PROPOSITION 3.2: Let X be a filtration on X* and let {Ln)n be a Cauchy séquence of languages over X (meaning that the séquence of closed subsets ofX* (jL„U{oo})n is Cauchyfor Dj). Then (Ln)n is a convergentséquence. Proof: Our hypothesis on (Ln)n is the following: Vn > 1, We may then choose an increasing séquence of integers (kn)n such that for all m > kn, we have Lm\In = Lkn\In. In particular, Lkn\In Ç i f e n + 1 \ / n + 1 . .Let now L — \J L^n\In. It is easy to verify that L \ / n = L^n\In for all n n and hence that L = ïim.jLn. D In the sequel we will use the following notion. We say that a filtration X = (ƒ„)„ is fast if Vw e X*, 3k > 0, Vn > 1, wn g Ik. Example: Let X = {a, 6}, let IQ = X\ and let In = X*an6X* for all, n > 1. Then Z = (/n)n is a fast rational ideal filtration. It is easy to verify Üiat 1 is a filtration. Suppose now that wn G Ik for some w G X*, n > 3 and k > 1. Then akb is a factor of wn, and hence a^6 is a factor of te2. Since Ç^In = 0, there exists fc such that u> and u>2 do not lie in /&. Then, for all \ > 1, wn .0 Ifc. 3.3. Séquences of automata We now consider automata (with unspecified set of final states) over X. The reader can consult Appendix 1 for a simpler version, stated for congruences instead of automata. Recall that we assume all automata to be accessible, and that A(X) is the class of all (accessible) automata over X (vee Section 1.1). To an automaton A — (Q, X, go, A) G A (X), we associate the équivalence relation CA = {(u, v) E X* x X*i% - u = qo •• v}. Note that, for ail u, v, w € X*, if (u, v) € CA, then (uw, vw) G CA. That is, GA is a right congruence of X*. A morphism tp from A = (Q, X, go? A) € -A (X) into A' — (Q', X, ÇQ, A') G v4 (X) is a mapping <p : Q -> Q7 such that, for ail u G X*, (A (go, «)) y? = A7 (gó, u). Thus', there exists a morphism from A to A; if and only if the équivalence relation CA refines CA>. In that case, the morphism from A to A1 is unique. Furthermore, since the automata we consider are accessible, the mapping tp is necessarily onto. If A = (Q, X, go, A) and A' = (Q\ X ' , qf0, À') are in X (X), we define Z)j (A, A') as follows (where A dénotes the symmetrie différence). Dj (A, A') = max {d (n, v)\(u} v) G CA A CA,} = max {d (u, v)\(qo - u =z go • u and q® • u ^ gó • t') or (go • u ^ g0 . v and gó * u = gÓ • Ü ) } . Let us remark that JDj (A, A7) = 0 if and only if CA — CA>, that is, if and only if, for ail u, v G X*, go * u = go • v if and only if gQ • u = g(j • v. Therefore, Dj (A, A') = 0 if and only if A and A' are isomorphic. In the next few lines we will verify that Dj is an ultrametric distance function on the set of isomorphism classes of automata over X. It would be equivalent to define Dj directly on the set of right congruences of X*. rational subgroup-cyclic) for all n. We say that a class of languages C is closed under bounded limits (resp. bounded rational limits, bounded rational subgroup-cyclic limits) if, whenever L = lim jLn and the limit is bounded (resp. bounded rational, bounded rational subgroup-cyclic) and Ln E C for each n, then L € C. THEOREM 4.1: For each integer k, jd$0T satisfies (BT) and (Br1) and is closed under limits. £btor satisfies (BT), (H"1), (Pr) and (Lpq), and it is closed under bounded Umits. THEOREM 4.2: For each integer jfc, £JJ satisfies (BT) and (H_i) and is closed under limits. £bap satisfies (BT), (H"1) and (Pr), and it is closed under bounded limits. Caution: It is important to note that, for any fixed k, £%t0Tdoes not satisfy (Pr and <Lpq), and that Ju^ does not satisfy (Pr). For instance, the product of two fc-bounded torsion (resp. aperiodic) languages is bounded torsion (resp. aperiodic), but its torsion bound may be greater than k. THEOREM 4 3 : Let k > 1, let k! = max (fc, 4) and L Ç X*, If L G £^^r (resp. L € J^bJp^ tnen t^iere exists a séquence (Ln)n and a f ast ideal filtration X = (ïn)n such that L — limjLn and Ln and In are rational (resp. star-free) and in £^JT (resp. JO^JV). The above results yield immediately the following descriptions of £btor and jCbap» which generalize the theorems of Kleene and Schützenberger. THEOREM 4.4: (Bounded generalizatioii of Kleene's theorem) £btor is the least family of languages containing the rational languages and closed under bounded limits. THEOREM 4.5: (Bounded generalization of Schützenberger's theorem) ^bap iiy Ihe leastfamily of languages containing the star-free languages and closed under bounded limits. Equivalently, A>ap is the least family of languages containing the finite languages andclosed under Boolean opérations, product and bounded limits. Remark: Theorems 4.1, 4.2 and 4.3 allow in fact more précise descriptions: A)tor (resp. £bap) is the least family of languages containing the rational (resp. star-free) languages and closed under bounded rational limits wifli respect to fast ideal bounded rational fibrations. THEOREM 4.7: (Bounded generalization of Straubing's theorem) £btor is the least family of languages containing the finite languages, satisfying (Lpq) and closed under Boolean opérations, product and bounded limits. Remark: As above, Theorem 4.6 implies in fact a more précise statement: A>tor is the least family of languages containing the rational subgroup-cyclic languages and closed under bounded rational subgroup-cyclic limits with respect to fast idéal bounded rational subgroup-cyclic filtrations. 4.2. Torsion and aperiodic languages In order to characterize the torsion and aperiodic languages in gênerai, it is not sufïïcient to consider only limits of séquences of languages. In fact, as we noticed in Section 3.2, every language is the limit of a séquence of finite languages modulo a cofinite filtration. A class of languages C is is closed under approximation (resp. under fast approximation) if, for each filtration (resp. fast filtration), X, whenever an automaton A is approximated modulo X by a séquence of automata (An)n such that C (An) Ç C for ail n, then C (A) Ç £. THEOREM 4.8: £tor satisfies (BT), (H"1), (Pr), (St) and (Lpq), and isclosed under fast approximation. THEOREM 4.9: £ a p satisfies (BT), (H"1) and (Pr), and is closed under fast approximation. THEOREM 4.10: Let A be an automaton over X such that C{A) Ç £ t o r (resp. C (A) Ç £ a p ) . Then there exists afast rational (resp. star-free) idéalfiltration X and a séquence of finite automata (An)n (resp. such that C (An) Ç £ a p for ail n), such that (An)n converges modulo X, (An)n approximates A modulo T, and the approximation is bounded aperiodic. As in Section 4.1, there results yield descriptions of £tor and £ a p which generalize Kleene's and Schützenberger's theorems. THEOREM 4.11: (Unbounded generalization of Kleene's theorem) Aor is the least family of languages containing the rational languages and closed under fast approximation, Equivalently, Aor is the least family of languages containing the finite languages and closed under union, product, star and fast approximation. THEOREM 4.12: (Unbounded generalization of Schützenberger's theorem) £ a p is the least family of languages containing the star-free languages and closed under fast approximation. Equivalently, £ a p is the least family of languages containing the finite languages and closed under Boolean opérations, product and fast approximation. Remark: As above, we can in fact make these statement more précise: Aor (resp. £ a p ) is the least family of languages containing the rational (resp. star-free) languages and closed under fast bounded aperiodic approximation with respect to a fast rational (resp. star-free) ideal filtration. Analogous to Theorem 4.6, we also have THEOREM 4.13: Let A be an automaton over X such that £ (A) Ç Aor (resp. C (A) Ç £ap)- Then there exists a fast rational subgroup-cyclic ideal filtration X and a séquence of finite automata (An)n such that the languages recognizedby the An are rational cyclic, (An)n converges modulo X, (An)n approximates A modulo X, and the approximation is aperiodic. THEOREM 4.14: (Unbounded generalization of Straubing's theorem) Aor is the least family of languages containing the rational subgroup-cyclic languages and closed under fast approximation. Equivalently, Aor is the least family of languages containing the finite languages, satisfying (Lpq) and closed under Boolean opérations, product and fast approximation. 4.3. Arbitrary languages The important Theorems 4.10 and 4.13 above are "fast' versions of results that hold for arbitrary automata, namely Theorems 4.15 and 4.16 below. Note that the latter differ from Theorems 4.10 and 4.13 only by the fact that the filtrations they involve need not be fast. As we will see in Section 5.2, the proofs of Theorems 4.10, 4.13, 4.15 and 4.16 are very similar. THEOREM 4.15: (Arbitrary language theorem) Let A be an automaton over X. Then there exists a rational idéal filtration X and a séquence offinite automata (An)n such that (An)n converges modulo X, (An)n approximates A modulo X, and the approximation is bounded aperiodic, THEOREM 4.16: (Cyclic arbitrary language theorem) Let A be an automaton over X. Then there exists a rational subgroup-cyclicidéal filtration X and a séquence offinite automata (An)n such that the languages recognized by the An are rational subgroup-cyclic, (An)n converges modulo X, (An)n approximates A modulo T, and the approximation is aperiodic. 5V PROOFS OF THE MAIN RESULTS The proofs of Theorems 4.1, 4.2, 4.8 and 4.9 can be obtained using more or less elassical methods, and we shall give these proofs in Section 5.1. As for Theorems 4.3, 4.6, 4.10, 4.13, 4.15 and 4.16, their proofs is more delicate and requires some deep results of semigroup theory. These proofs will be discussed in Section 5.2. Finally, we note that given these theorems, the bounded and unbounded generalizations of Kleene's, Schützenberger's and Straubing's theorems (Theorems 4.4, 4.5, 4.7, 4.11, 4.12 and 4.14) are immédiate. 5.1. Proof of Theorems 4.1,. 4.2, 4.8 and 4.9 First we observe that rational languages are trivially in A>tor and that aperiodic (star-free) rational languages are in A>ap The fact that the families £$ot> 4 Ï P ' £btor, 4>aP. Aor and £ a p satisfy (BT) is a direct conséquence of Proposition 1.3, together with Lemma 1.2 and Proposition 2.3. The same Lemma 1.2 and Proposition 2.3 show that these classes of languages satisfy (H"1). Let us now show that they ail satisfy (Pr), and that Aor and A>tor satisfy (Lpq). The proof uses the properties of the Schützenberger product and of its "mod ^''-variant. For more detail on these products, see in particular [15, 22, 23]. PROPOSITION 5.1: Ator, A a p , Aor and C^ satisfy (Pr). Proof: Let L and L' be languages in X*, let S (L) and S (1/) be their syntactic monoids, and let 77^ and r\y be their syntactic morphisms. Let rj Informatique théorique et Appliçations/Theoretical Informaties and Applications be the morphism from X* into the multiplicative monoid of (2,2)-matrices with entries in the semiring (V S (L) x S (L1)), U, •) defined by One can check that, for all w E X*, we have where Pw — {(U7]L7 vq.L')\w. = uv}. Then, LU — ifry"1, where K is the set of all matrices in the form with P H (LT7^ x Ur]L') ^ 0. The range of 77 is denoted X*r] = O2 ( S ( i ) , 5 (£')) and i s c a l l e d t h e Schützenberger product of S (L) and S (L''}. Let now TTbe the morphism from O2 (S (L% S (U)) into S(L) x 5 (LA) defined by WTJIT = {WTJL, wr]z>). Let e and e' be idempotents, respectively in S (L) and S (U). Then the inverse image (e, e') TT"1 satisfies the identity x3 = £4. Let indeed m G (e, e7) TT"1, say, 0 {(! Then, for each n > 1, m71 is in the form where Pi = P and Pn+\ = (e, 1) P U Pn (1, e'). In particular P2 = (e, 1 ) F U P ( 1 , e') P3 = (e, 1}P U (e, 1) P (1, e') U P (1, e') P n = P 3 / o r aH n > 3. Therefore TT is a bounded aperiodic morphism and henee (see Proposition 2.2), if S (L) and 5 (L') are in Tor, bTor, Ap or bAp, then so are 0 2 (5 (L), S (L')) and S (LI/). D vol. 29, n° I, 1995 2 8 PROPOSITION 5.2: A>tor &nd Aor satisfy (Lpq). Proof: The proof is quite similar to the proof of Proposition 5.1. Let p be a prime number, let L be a language in X*, and let T]L : X* —»5 (L) be its syntactic morphism. Let r\ be the morphism from X* into the multiplicative monoid of (2,2)-matrices with entries in the semiring ïp [S (L)} defined by xr]=(x^L M for ail x G X. Zp [S (L)] is the semiring of formai linear combinations of éléments of S (L) with coefficients in the cyclic group of order p, Zp. (Note that the semiring {V (S), U, -) can be identified with B [S] where B is the Boolean semiring.) One can check that, for ail w G X*, we have 's P 0 1 e m = [(0 p) 1) with Pw — 2_\ur)L where the sum runs over all préfixes u of w. Then 77 recognizes (L, p, q). Indeed, if K is the set of matrices of the form with P = \ ^ css with \ ^ cs = q (mod p), then (L, p, g) = s€S(L) seLr)L The range of 77 is denoted X*r) — Zp^2 {S (L), 1). Let now TTbe the morphism from Zp O2 (S (L), 1) into 5 (L) defined by WTJ-K = WTJL. Let e be idempotent of S (L) and let m G eTr"1, that is, Krj'1. Then, one vérifies by induction that for each n > 1, Therefore mpJrl = ?Ti and hence TTis a bounded torsion morphism. By Proposition 2.2, if S (L) is torsion (resp. bounded torsion), then so are ZVO2{S{L), 1) and 5 ({L, p, g)). D w = WQai w\ ... a*; Wk with w% £ Li for 0 < i < k is congruent to r modulo n. This product was studied in detail in [22, 23], and the above proof shows in fact that A>tor and Ct01 are closed under such products. Our next task will be to prove the following result. Our proof extends ideas contained in the proof of a less gênerai result, due to Straubing [21]. PROPOSITION 5.3: Aor satisfies (St). Proof: Let L Ç X* be a torsion language and let TJL : X* —• S (L) be its syntactic morphism. Let w G X*. Then there exists n > 1 such that wnr]L = w2nr}L> Let x = itA Then XT)L = z2VL' It will be sufficient to prove that there exist k and kf > 1 such that xkrqi* = xk+kfr}L*. Let fei = 2 + 4|s| and fe2 = (fe2 - 2)!. We will first prove that, for all u, v e X\ uxklv uxklv n ^ 1 + f c 2 i ; G L*, 3 1 < t < fei - 2, u x f c l - ^ G L*. Indeed, if uxklv G L*, then uxklv = xi . . . xp for some r e i , . . . , x^ G L. For each 0 < r < 1 + 2|x| = fei/2, let ir = min {1 < i < p\ux r is a prefix of xi . . . x2}. Then r H i r is a non-decreasing function from { 0 , . . . , 1 + 2|x|} into { 1 , . . . , p}- If the function r H ir is not injective, then there exists 0 < r < 2\x\ such that ir — i r + i , that is, ux2r — xi . . . Xir_1u/ and #1 . . . XiT — ux2 (r+1)'u/ for some non-empty prefix v! of x%r and for some vl G X*. In particular Xir = ufx2v' and^'x^+i . . . xp = x f c l ~ 2 ( r + 1 ^ . Since #77L = ^2Ï?L = xtr]L fof all t > 1, we have vfxv' G L and u/x2+tv/ G L for all * > 1. Therefore utf*1"-1*; = ux2rxxkl~2 ( r + 1 ) i ; = xi . . . ^ r _ i ( u ' W ) xir+i .. . xp <E L* and, for t > 1 tv = nx2rx2^x^-2^^v — xi . . . Xir-i (u'x2+V) Xir+\ ... xp e L* Now, if the function r H-> zr is injective, then for each 0 < r < 2|x|, we have xi . . . x;r •= u x 2 r ^ i (r) xp = w2 (r) ^i-2{^+i) v for some words î^i(r) and-tü2{r) such that wi{r)w2'(r) —x2 andtt;2(^) ¥" 1Since r can assume 2\x\+ 1 values, there exists 0 < r < rf < 2\x\ such that w\ (r) —w\ (r') and W2 (r) = W2 (r'). Therefore = x i . . . Xir Xir,+i . . . x p GL*, and, since x ^ + i . . . Xir, — W2 (r) x2 (r>/-r~1) Wl (y^ for ail t > 1 we have ^2 (r'-r-l) = xi . . . xir ( x ^ + i . . . xir, )f xirt+1 xp E L*. Note that 2 (r' - r) < 2r' < 4jx| < fci - 2. Therefore, in ail cases and if k2 = (h - 2)!, uxklv E L* => ux*1+fev G L*, ÎXX^1^ G L* =£>uxkl~fv G L* for some 1 < i < fci - 2. Let now y = x^2 and A;3 = 2 + 4]y|. Since y ^ = y27]L, the above computation shows that uykzv G i * ^> uykz~fv G L* for some 1 < t < h ~ 2, that is, uxk2ksv G L* => itar^3"*)*2?; G L* for some 1 < t < fc3 - 2. Informatique théorique et Applications/Theoretical Informaties andApplications 3 1 Now, (k$ —t) &2 > 2% > k\, so that ux(k3-t)Jc2y eL* k*+***v E V for all s > 1 and hence Conversely, since &2% ~ &2 > &i> uxk*k*v E V Therefore xk2k3~~k2r}£* = xk<2k3r]^, which concludes the proof. (k) The following proposition will help us show that ^tov closed under limits. D are PROPOSITION 5.4: Let X = (In)n be a filtration of X* and let L Ç X* be the limit L = lïmjLn for some séquence of languages {Ln)n. Then S (L) <Y[S {Ln) n Proof: Let r}n : X* —>5' (Ln) be the syntactic morphism of Ln and let 77 : X* —> TT S (Ln) be defined by urj — (ur]n)n for all ti 6 X*. By n Lemma 1.2, it suffices to show that r\ recognizes L. Let us assume that u, v E X* are such that wq — vq and u E L. Since [ j In — 0, there exists no > 1 such that u and v are not in ƒ„.. Since L = I i m l n , there exists ni such that L\Ino — Lni\In0uï]ni — V7ini, so v E Lni\Ino In particular, u E Lni. But UT/— vq implies and hence v E L. Thus 77 recognizes L. • CoROLLARY 5.5: For «// k, the classes C^tJ0T and JC{,^P « ^ closed under limits. In particular, £btor a w ^ ^bap a ^ closed under bounded limits. Proof: We known that the classes bTor^ and bAP/. are closed under arbitrary direct product and taking finitely generated divisors. In particular, if I is a filtration of X* and if L = MmL^ where the L n are all in ^l (resp. jCjj.Jp), then so is L. This shows immediately that the classes and £bap satisfy the required closure property. D To complete this section we prove the following important proposition. PROPOSITION 5.6: Let £tor and £ap are closed underfast approximation. vol. 29, nQ 1, 1995 3 2 Proof: Let X = (ƒ„)„ be a fast fîltration, and let A = (Q, X, g0) A) and An = (Qn, X, ç j , Àn) (n > 0) be automata such that C (An) Ç £ t o r for ail n and such that the séquence (An)n approximates A modulo X. By Proposition 2.5, we known that 5 (An) G Tor for each n, and we need to show that S (A) e Tor. Let u G X*. Since X is a fast fîltration, there exists an integer n such that n* H In — 0. Let now kn be the integer associated to n in the définition of approximation. Since S (An) G Tor, there exists r > 1 such that q-ur = ç - ^ 2 r for ail g G Qfc^- This is equivalent to ÇQ™ • W = qon -vu2r for ail u G X*. Because of the way we chose fcn, this implies go * ^^r = Qo •^^2r for ail v G X*, and hence q • uT = q • u2r for ail q G Q. Thus 5 (A) is torsion and hence C (A) Ç £tor The proof of the statement concerning £ a p is similar. D 5.2. Proof of Theorems 4.3, 4.6, 4.10, 4.13, 4.15 and 4.16 These proofs require three deep results from semigroup theory. The first of these results (Theorem 5.7 below) is a property of finitely generated Burnside monoids, defined by identities of the form xk — xkJrl. This resuit was first proved by McCammond [11] for k > 6, and extended to k > 4 by Pereira do Lago [14] using différent methods. The two other results are properties of the semigroup expansions S \—> S^ and S H-> S (cut-down to generators) considered by Rhodes and others. A complete study of these expansions, including the proof of the results stated hère, can be found in the works of Birget, Henckell, Lazarus and Rhodes [1, 17, 18, 6]. See also Appendix 2 below. THEOREM 5.7: (See [11, 14]) Let X be afinite alphabet, and let k and l be integers with k > 4 and l > 1. Let Bx (&, l) be the monoid generated by X and defined by the identity xk = xk+l. (1) The maximal subgroups of Bx (k, l) are finite cyclic groups. (2) There exists a séquence (Jn)n of ideals of Bx (&, 0 such that O Jn = 0s and Bx (k, Ï)/Jn is afinite monoid. n THEOREM 5.8: (See [1]) Let S be a finitely generated semigroup, and let a : X* —» S be an onto morphism, with X a finite alphabet There exists a semigroups S^\ an onto morphism r : X* —• S^ TT : S^ —• S which satisfy the following property and a morphism (1) T7T = (7. (3) 77ie subgroups of S^ ( 4 ) There exists a séquence (Jn)n of ideals of S^ such that (2) 7T « <2 5-bounded-aperiodic morphism. are isomorphic to the finite subgroups of S. = Jo D Ji D . . . D J„ D . . . n is a finite monoid. THEOREM 5.9: (See [17, 18]) Let S be afinitely generated semigroup, and let G : X* —>5 be an onto morphism, with X a finite alphabet. There exists a semigroup S, an onto morphism a : X* —>- 5 and a morphism ip : 5 —> 5 which satisjy the following property (1) <pil) = a. (2) ij) is an aperiodic morphism. (3) The maximal subgroups of S are finite cyclic groups. We are now ready to give the remaining proofs. Proof of Theorems 4.3 and 4.6: Let L Ç X* and let a : X* -> S be its syntactic morphism. If L E £btor> then S G bTor, and hence 5 G bTor^ for some integer k. Since k <k\ S belongs to bTor&/ as well. Let [i be the canonical morphism from X* onto Bx (kf, fc'). Then there exists an onto morphism y> : Bx (k*, fc') —>*S such that //y? = a. Let (Jn)n be given by Theorem 5.7 and let In — JnjjT1 for each n > 0. J — ( / n ) n is trivially an idéal filtration on X*. Furthermore, each In is recognized by the morphism \in : X* —• S x (fc7, kf)/Jn obtained by composing fi with the canonical projection of Bx(kf, k!) onto Bx(k' ,k!)/Jn. This proves that In is rational subgroup-cyclic and fc-bounded-torsion. The fact that Bx (kf, /e7) is torsion also implies that Z is a fast filtration. Indeed for each w G X*, the set w*fi is finite, so there exists n > 1 such that 1^*// n J n = 0 and hence w* D Jn = 0. Let now Ln = L\In for each n > 0. Since L is recognized by <r, it is recognized by /x too, and hence Ln is recognized by the morphism /%, so that Ln is rational subgroup-cyclic and Â/-bounded-torsion for all n. It is then easy to verify that L = lim jLn and the limit is bounded rational subgroup-cyclic. In the case where L G £{J , we have 5 G bAp&, and we consider (kf, 1) instead of Bx (AA7;7, A:'). The corresponding statement of Theorem 4.3 follows since Bx (&', 1) G bAp^/ by Theorem 5.7. D Proof of Theorem 4.15: Let A = (Q, X, </ch A) be an automaton and let a : X* —» 5 = S (A) be its transition morphism. Let also r : X* -» 5(3), 7T : 5(3) —> 5 and (J n ) n be given by Theorem 5.8 and let In — JnT~~l for each n > 0. As in the above proof, one vérifies that, for each n, In is recognized by the onto morphism r n : X* —» S^/Jn, obtained by composing r with the projection of 5^3) onto S^/Jn. Note that S^/Jn is finite. So X = (/n)fi is a rational idéal filtration. For each n, let An = A (S^/Jn) be the automaton associated with the morphism rn, It is not difficult to verify that the séquence (An)n converges to A ^ modulo X. Furthermore Q UT \-+ qo - u is a well-defined morphism from A (S^3)) to A, so that the séquence (An)n approximates A modulo 1 (Proposition 3.5). There remains to verify that the approximation is bounded aperiodic. Let u G X* be such that q • u = q • u2 for ail q G Q. Then u a = u2a and hence u^r = u6r since 7T : S^ —• .5 is 5-bounded aperiodic. Therefore, if s is any state of A(5(3)), then s * u^ = s (ubr) = s (u6r) —s -u6. This concludes the proof. D Proof of Theorem 4.10: The proof of Theorem 4.15 given above can be copied Verbatim. We only need to verify that the filtration I is fast, and that the In and the languages of the An are star-free if C (A) Ç £ a p . Since n is 5-bounded-aperiodic, S^3) is torsion (resp. aperiodic) if 5 is, that is, if C (A) consists only of torsion (resp. aperiodic) languages. In particular, if L (A) C £ a p , then the monoids S^/Jn are finite and aperiodic, so that In and the languages of An are star-free. The proof that 1 is fast is as in the proof of Theorem 4.3. • Proof of Theorem 4.16: This proof is quite similar to the proof of Theorem 4.15. Let A — (Q, X) qo, X) be an automaton and let a : X* -^ S = S (A) be its transition morphism. Let <p : X* —*•S and ip : S —»• S be given by Theorem 5.9. We now apply Theorem 5.8 to the morphism <p. Let T — S { \ We obtain onto morphism r : X* —>T, TT : T —> S and a séquence (Jn)n of ideals of 'T. Let now X = (In)n with In — JnT~x for all n > 0. Reasoning as in the proof of Theorem 4.15, we verify that each In is recognized by a morphism rn : X* —> T/Jn, which proves that In is rational subgroup-cyclic. Again, we verify that, if An — A (T/Jn), the séquence (An)n converges to A (T) modulo I, and there exists a torsion morphism from A (T) to A Thus {An)n approximates A modulo X and the approximation is aperiodic. D Proof of Theorem 4.13: We only need to complete the proof of Theorem 4.16 with the conséquences of the fact that C (A) C Aor» that is, S E Tor, and this is done as in the proof of Theorem 4.10. O Appendix 1: Séquences of morphisms Instead of considering séquences of automata as we did in Section 3.3, we could have considered of X-generated monoids, or more precisely, of onto morphisms defined on X*. To such a morphism a : X* —> S we associate the congruence Ca = {(u, v) € X* x X*\ua = va}. Given a filtration X on X*, we then define a distance function Dj by Dj (O"J T) = max {d (w, ^)|(^, v) G CaACT}. As in Section 3.3, D is in fact a distance function on the set of isomorphism classes of onto morphisms defined on X*, which is ultrametric. Results analogous to Propositions 3.3 to 3.5 hold for this metric too. In particular, one can define the concept of a séquence of onto morphisms approximating an onto morphism modulo a filtration, and the concept of a class of monoids being closed under (fast) approximation. Within this framework, statements similar to those of Sections 4.2 and 4.3 hold, and they are obtained with proofs quite similar to those reported for Proposition 5.6 and in Section 5.2. Appendix 2: Rational filtration associated to an arbitrary congruence Let = be an arbitrary congruence on X* and let er : X* —> S — X* / = be the canonical projection morphism. In this section we will show how a rational ideal filtration is naturally associated with = . Let us emphasize that ^ is arbitrary but that the (simple) objects which we construct from =-computations are rational! Recall the définition of Green's relation J. If s and t are éléments of a monoid S, we say that s < jt if SsS Ç StS and that sjt if SsS = StS. Then J is an équivalence relation and < ^ is a quasi-order. If 5 < jt and no£ {sjt), we write 5 < j - £. We say that a word n i s a factor of a word v if i> = xm/ for some x, ?/ G X*. This is equivalent to saying that v < ju in the ^7-order of X*. Let X= = (In)n be the séquence of subsets of X* defined for each n > 0 by In — {w G X*||{ucr|tiis a factor of it;}| > n } . Sketch ofthe proof of Theorem 5.8: Let Vf (Ss) be the set of finite subsets of 5 3 = S x S x S. For (51, 52, 53) and (54, 55, 56) in S3 we define ( 5 1 , 52, S 3 ) * ( S 4 , 5 5 ' S 6 ) =< with empty products equal to 1 by convention. This opération is extended to Vf (S3) by A * B = {a * 6|a G A, 6 G 5 } for all A, B eVf (S3), which makes £>/ (S3) a monoid. Moreover, letting WT = for each w G X* defines a morphism from X* into Vf (S3). Let 5<3) = X V . For each (si, S2, 53) in Vf (S3), let (si, S2, 53)7r = 51^253. Then a = r7r. Next one vérifies that if w G X* and tca is an idempotent of S, then iu5r = w6r. The argument is standard: if w6 is factored as w6 — then one of w\, W2 and w$ contains w2 as as factor. This shows thatTT is bounded aperiodic. For each x G S^\ let set (x) and F (x) be the sets set (x) = {s G S|3si, 52 G 5, (si, s, 52) G #} and Then set (#) is a finite set, containing ZTT, and x < j-ç/ implies set (y) Ç set (#). Therefore, for each fixed x, the set {set (y)\y G F (x)} is finite, and hence F (x) is finite. By elementary facts of semigroup theory, if follows, that each strictly ascending chain in the <j -order is finite, each ^-class of S^ is finite, and so is each of its subgroups. For each n > 1 let us now define t ( x ) | > n}. Since x <jy implies set (y) Ç set (x)9 Jn is an idéal, and Pj Jn = 0. We n now want to prove that S^\Jn is finite for all n. Since Jo = S^ it will be sufficient to show that is finite for all n. Note that Jo is a finitely generated semigroup. We will prove by induction on n that Jn\Jn+i is finite and Jn+1 is finitely generated (as a semigroup) for all n > 0. If Jn is generated by a finite set Xn Ç Jn and if x — x\ . . . Xk (x{ G Xn), then we have set (xi) Ç set (x) for all i. Thus |set (x)\ — n implies set (x) = set (XJ) for all 1 < i < fe. Therçfore Jn\ Jn+i is contained in the union of the sets set"1 (set (x)) when x G Xn and hence is finite. Let now Yn = {rx\r G Jn\Jn+i U {1}, x G l n and rx G J^+i} and Xn+i = Yn\JYn ( J n \ J n + i ) . Then X n + i is finite and, since J n + i is an idéal, Xn+i is contained in J n + i . Let now y G J n + i - Then y — x\ .. ,xjz for some a ; i , . . . , a;^ G X n . Let fci > 0 be minimal such that x\ . . . Xkx+i G Jn+i. Then r\ = x\ .. . x ^ G Jn\</n+i and Iterating this reasoning we obtain a factorization with riarfcl+i e Yn,...., n r r ^ + i in Yn and rÉ+i G J n \ J n + i U {1}. Thus Xn+i is a finite System of generators for J n + i . D Remark: Using the notations of the above proof, let We could prove in a similar fashion that (Kn)n is another séquence of ideals of S^ satisfying condition ( 4 ) of Theorem 5.8. Proof of Theorem A.2.1: We use the notations of the above proof. Let rn : X* -> S^\Jn be the composition of the morphism r : X*' —> S^ with the projection of S^ onto S^/Jn. For tu G X*, note that set (WT) — {ua\u is a factor of w}. Therefore In = Jnr~1, so that In is an idéal and In is recognized by the morphism rra. Since S^/Jn is finite, 7n is rational. D Appendix 3: Generalization of Eilenberg's variety theorem In this section we give a generalization of Eilenberg's variety (or stream) theorem (see [4, 16}) which encompasses the classes of (bounded) torsion and aperiodic languages. Recall that for us, a language is always a subset of some free monoid X* where the alphabet X is finite. We say that a monoid S in syntactic if it is the syntactic monoid of some language. In particular S is necessarily finitely generated. If P is a subset of a monoid S, we say that P is disjunctive if for ail 5, t G S (Vu, v G 5, usv e P & utv e P) s = t. (In other words, the syntactic congruence of P in 5 is trivial.) The following resuit is well-known. LEMMA A.3.1: Let S be a finitely generated monoid. Then S is syntactic if and only if S contains a disjunctive subset. This lemma allows the description of a large class of syntactic monoids. Informatique théorique et Applieations/Theoretical Informaties and Applications ALGEBRAIC ANDTOPOLOGICAL THEORY OFLANGUAGES PROPOSITION A.3.2: Let S be afinitely generated semigroup. Let usassume that S admits a séquence of ideals (Jn)n such that Jn — 0, and, for all n > 0 and for all pairs of distinct éléments of S, n s ^ t, there exists u, v E S such that usv 7^ utv and usv, utv E Jn Then S is a syntactic monoid. Proof:For each x e 5, let r (x) = min {n > Q\x g Jn}- Notice that S is countable since it is finitely generated. Let be an enumeration of the pairs of distinct éléments of S. We now construct by induction two séquences (un)n and (vn)n of éléments of S such that, for each n, unsnvn / untnvn and {r (unsnvn), r (untnvn)} > max{r (uiSiVi), r (u.itiVi)\0 < i < n}. By hypothesis, there exist UQ and VQ such that UQSQVQ ^ uotovo- Let us now assume that n o , . . . , um have been chosen which satisfy (*) for all n < m. Let Tm — max {r (uiSiVi), r (uitiVi)\0 < i < m}, By hypothesis, there exists n m + i and vm+i such that n m + i s m + i t ; m + i ^ % + i t m + i % + i a n d u m + i % + i % + i , % + i W i % + i € Jr» that is, r ( r t T O + i s m + i v m + l ) > ^m and r (um+itm+ivm+i) > rm. For these values of w r a + i and w m + i , (*) is again satisfied. Let now F — {unsnvn\n > 0 } . We claim that P is a dïsjunctive subset of 5, and hence that S is syntactic by Lemma A.3.1. Indeed, if s ^ t C 5 , then 5 = sn and t — tn for some n. Then unsvn •— unsnvn E F . Furthermore, i4TCtun •= uninvn is not equal to unsvn, and for all i <n< jf, we have r (uiSiVi)< r (untvn) < r (UJSJVJ), so that untvn g P. D COROLLARY A.3.3: If X is afinite alphabet, then X* is syntactic. Proof: It is easy to verify that X* satisfies the hypothesis of Proposition A.3.2 for Jn = {w G X*\\w\ > n} (n > 0). El Let V be a class of finitely generated monoids. We say that V is a variety of finitely generated monoids, or fg-variety, if: (1) V is not empty. (2) If Si, S2 G V, then Si x S2 G V. (3) If S G V , if T is finitely generated and if T divides S, then T G V , ( 4 ) For each S G V , there exists a finite collection S i , . . . , Sn of monoids in V which are syntactic and such that S < Si x . . . x Sn. Remark: We could also define an gf-variety by properties (l)-(3), and then restrict ourselves to fg-varieties which are generated by a class of syntactic monoids. Example: Let FG be the class of ail finitely generated monoids. Then FG is an fg-variety. Indeed, by Corollary A.3.3., each finitely generated monoid is a quotient of a syntactic monoid. The usual varieties of finite monoids, or M-varieties are exactly the fg-varieties consisting only of finite monoids. Let £ be a class of languages. We say that £ is a variety of languages if: (1) £ is not empty. (2) £ satisfies (H"1). (3) £ satisfies (BT). Example: The class £an of ail languages is a variety. The resuit of Section 4 show that, for k > 1, ^ t o r ' ^bap> A>tor> A a p . Aor and £ a p are varieties of languages. Also, the usual varieties of rational languages are exactly the varieties of languages consisting only of rational languages. Let V be an fg-variety. We define £ (V) to be the class of all languages whose syntactic monoid is in V, or equivalently, the class of all languages that are recognized by some monoid in V. Also, it if £ is a class of languages, we let V (£) be the fg-variety generated by the syntactic monoids of the languages of £. With these notations, we can state the following generalization of Eilenberg's variety theorem. THEOREM A.3.4: The correspondence V H-> £ (V) is one-to-one and onto from the class of ail fg-varieties onto the class of all varieties of languages. Furthermore the reciprocal correspondence is given by £ i—• V (£). Proof: The proof is very similar to that of Eilenberg's variety theorem. Let V be an fg-variety. We verify that £ (V) is a variety of languages by an immédiate application of Lemma 1.2 and Proposition 1.3. It is clear that, if V and W are fg-varieties and V Ç W , then £> (V) Ç £ ( W ) . Let us prove that the converse holds, that is, that £ (V) Ç £ (W) implies V Ç W . Note that this will prove that the correspondance V H £ (V) is one-to-one. Let S E V. By définition of an fg-variety, we have S < Si x . . . x Sn where Si E V and Si is the syntactic monoid of some language Li. Then each Li is in £ (V) and hence in £ ( W ) . Therefore Si E W for each i and hence S E W . Finally let £ be a variety of languages. We will show that £ (V (£)) = £ , thus showing that the correspondences V H £ (V) and £ H-> £ (V) are mutually reciprocal. The inclusion £ Ç £ (V (£)) is trivial. To prove the converse we consider L Ç X* with L ^ C (Y (£)). We know that the syntactic monoid of L, S = S (L), is in V (£) and hence that there exist finitely many languages Li Ç X * , . . . , L n Ç X* such that L i , . . . , Ln E £ and S divides Si x . . . x Sn (where Si is the syntactic monoid of Li). Let 771,..., 77^ be the syntactic morphisms of L i , . . . , Ln and let e be n a symbol not in M Xj. For each 1 < i < n we define a morphism (j% : (X, U {e})* - ^ X* by xai = x for all x E X; and ea% = 1. Let y = ( X i U { e } ) x . . . x ( X n U { e } ) . Then a — (a\,..., an) is a morphism from Y* onto X^ x . . . x X*. Finally n n let 77— (771,..., rjn). Then 77 is an onto morphisms from T T ^ * o n t o TT^* Now S divides Si x . . . x Sn, so L is recognized by Si x . . . x Sn. That is, there exists a morphism ip : X* —• Si x . . . x Sn and a subset P such that L = Pip~l. Since ar\ is onto there exists a morphism n : X* —• Y* such that (p — 7T(ar}). Therefore L — P (arj)~17r~1 and it suffices to show that P (ar})~1 E £. The situation is summarized by the following commutative diagram where ni and n\ are the i-th projections. y* «-T x* 1 ^ (bl X . . . X *by^ Öi In particular T T ^ = 777^ for ail 1 < i < n. Let P2 = LiT\i and let LJ^ ^ t * ^ l X . . . X )Ji ^ X Ji<i X o ^ j - l X . . . X tJfi) Tj (T x . . . X * _ x x L , x X * + 1 x . . . x Note that L\ — Pi (cq^) 1 so that arjir^ recognizes L%* We show that arjTTj — cnviTji is in fact the syntactic morphism of L\. Let u, v G Y* be syntactically equivalent (for L[). Then for ail x, y G y * , (#uy) c ^ = (xuy) cnrirji G Pi if and only if (rru?/) CT^TT^ = (XVÎ/) CTTT^Ï G P^. Therefore (xuy) <J7Ti G Li if and only if (xvy) <77Ti G L2, that is, ua%ir\i — vair^i and hence uarjir^ = vaitirn. By Proposition 1.3, P Or?)"1 G 5 r ( L i , . . . , L'n). But Lj = L ; ^ ' " 1 , so L^ G £ and hence P (orr/)"1 G C. • Example: We already remarked that the coirespondences between Mvarieties and varieties of rational languages (see Pin [16]) are instances of the correspondence described in Theorem A.3.4. Other examples are given in the following thereom. THEOREM A.3.5. —Let k be an integer with k > 6. The classes FG, FG H bTorfc, FG n bAPfcî FG n bTor, FG n bAP, FG H Tor and F G n A p are fg-varieties. We have the following correspondences: (k) Kfcrl VIT»TA 1 i. /*^bt o'r b T o r H^ £ b t o r -1 * r* A n Ï l ^ / ^ ^ ~\k] oap b A p ^ £ b a p F G ^> £an . We will not prove this fact. Let us just say that readers familiar with McCammond [11] can use the notion of rank to prove that Bx (&, t)). satisfies the hypothesis of Proposition A.3.2., andhence is syntactic. Similar, in order to prove that F Gn Tor and F G n A P arefg-varieties, it sufflces to show that^for all 5" G F G n b T o r (resp. S € F G n A P ) ; then 5 is syntactic. Note that 5 € F Gn b T o r (resp. F GD-bAP) since the morphism ij) of Theorem 5.9 is aperiodic. Again, we leave it to readers familiar with Rhodes [17]to prove that S satisfies the hypothesis of Proposition A.3.2. This can be done using the natural filtration given by the length of the infinité itération matrix semigroup (IIMS) description of S. D REFERENCES 1. J.-C. BIRGET and J. RHODES, Almost finite expansions, Journ. Pure Appl. Alg., 1984, 32, pp.239-287. 2. A. DE LUCA and S. VARRICCHIO, On non-counting regular classes, in Automata, languages and programming (M.S. Patersen, éd.), Lecture Notes in Computer Science, 1990, 443, Springer, pp.74-87. 3. A. DE LUCA and S. VARRICCHIO, On non-counting regular classes, Theoret Comp. Science, 1992,100, pp. 67-104. 4. S. EILENBERG, Automata, languages and machines, vol.B, Academie Press, New York, 1976. 5. R. GRIGORCHUK, Degrees of growth of finitely generated groups, and the theory of invariant means, Math. USSR Izvestyia, 1985, 25, pp.259-300. (English translation AMS.) 6. K. HENCKELL, S. LAZARUS and J. RHODES, Prime décomposition theorem for arbitrary semigroups: gênerai holonomy décomposition and synthesis theorem, Journ, Pure Appl Alg., 1988, 55, pp. 127-172. 7. I. HERSTEIN, Noncommutative rings, Carus Mathematical Association of America, 1968. Mathematical Monographs 15, 8. J. HOWIE, An introduction to semigroup theory, London, Academie Press, 1976. 9. S. KLEENE, Représentation of events in nerve nets and finite automata, in Automata Studies (Shannon and McCarthy eds), Princeton, Princeton University Press, 1954, pp. 3-51. 10. G. LALLEMENT, Semigroups andcombinatorial applications, New York, Wiley, 1979. 11. J. MCCAMMOND, Thesolution to the word problem for the relatively free semigroups satisfying ta = ta+b with a > 6, Intern. Journ. Algebra Comput. 1, 1991, pp. 1-32. 12. J. L. MENICKE, Burnside groups, Lecture Notes in Mathematics 806, 1980, Springer. 13. E. F. MOORE, Sequential machines, Addison-Wesley, 1964, Reading, Mass. 14. A. PEREIRA DO LAGO, On theBurnside semigroups xn - xn^rn,. éd.), Lecture Notes in Computer Sciences, 583, springer. LATIN 92 (I. Simon 15. J.-E.PIN, Concaténation hiérarchies and decidability results, in Combinatorics on words: progress and perspectives (L. Cummings, éd.), NewYork, Academie Press, 1983, pp. 195-228. 16. J.-E.PIN, Variétés de langages formels, Paris Masson, 1984, (English translation: Varieties of formai languages, Plenum (New York, 1986). 17. J. RHODES, Infinité itération of matrix semigroups, I, J. Algebra, 1986, 98, pp. 422451. 18. J. RHODES, Infinité itération of matrix semigroups, II, /. Algebra, 1986, 100, pp. 25-137. 19. M.-P. SCHÜTZENBERGER, On finite monoids having only trivial subgroups, Information and Control, 1965, 8, pp. 190-194. THEOREM 4 . 6: Let k > 1, let kf = max (fc, 4) and L Ç X*. If L E £ ^ r , then there exists a séquence (Ln)n and afast idéal filtration X = (In)n such that L -limjLn and Ln and In are rational subgroup-cyclic and in £t>tor. Remark: Let LQ , L i , . . . , L & Ç P , a i , . . . , a& G X and r > 0, n > 2. We define the product with counter (LQai L\ ... a& L&) r, n to be the set of all words w G X* for which the number of factorizations in the form 20 . H. STRAUBING, Families of recognizable sets corresponding to certain varieties of finite monoids, Journ . Pure Appi Alg., 1979 , 75 , pp. 305 - 318 . 21. H. STRAUBING, Relational morphisms and opérations on recognizable sets , RAIRO Inform. Théor. , 1981 , 75 , pp. 149 - 159 . 22. P. WEIL, Products of languages with counter , Theoret Comp. Science , 1990 , 76 , pp. 251 - 260 . 23. P. WEIL, Closure of varieties of languages under products with counter , Journ. Comp. System and Sciences , 1992 , 45 , pp. 316 - 339 .


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J. Rhodes, P. Weil. Algebraic and topological theory of languages, RAIRO - Theoretical Informatics and Applications, 1995, 1-44, DOI: 10.1051/ita/1995290100011