A negtive nswer to question of Wilke on vrieties of!-lnguges Jen-Eric Pin () Astrct. In recent pper, Wilke sked whether the oolen comintions of!-lnguges of the form! L, for L in given +-vriety of lnguges, form the!-prt of n 1-vriety. We provide negtive nswer to this question. Vrieties were introduced y Eilenerg in 1976 to give unied frmework to the lgeric theory of recognizle lnguges. Recll tht +- vriety ssocites to every lphet A set A + V of recognizle lnguges of A + stisfying the following properties: (1) for ech lphet A, A + V is closed under nite union nd complement, (2) for ech morphism ' : A +! B +, L 2 B + V implies L' 1 2 A + V, (3) if X 2 A + V, then, for ll u 2 A, u 1 X; Xu 1 2 A + V. Eilenerg's well-known vriety theorem sttes tht +-vrieties re in oneto-one correspondence with vrieties of nite semigroups, tht is, clsses of nite semigroups closed under tking susemigroups, quotients nd nite direct products. This result is such powerful tool for clssyfying recognizle lnguges tht it ws nturl to try to extend it to!-lnguges. After some pioneering work y Perrin [3] nd Pecuchet [1,2], the right denition ws given y Wilke [4,6]. It turns out tht it does not suce to work only with innite words, ut tht nite words hve to e considered t the sme time. More precisely, when A is nite lphet, denote y A! the set of innite words on A, nd set A 1 = A + [ A!. A suset X of A 1 is identied with the pir (X + ; X! ) = (X \A + ; X \A! ). In prticulr, X is sid to e recognizle if oth X + nd X! re. An extension of the usul notion of quotients is in () LITP/IBP, Universite Pris VI et CNRS, Tour 55-56, 2 plce Jussieu, 75251 Pris Cedex 05, FRANCE. e-mil: pin@litp.ip.fr
2 order for the min denition. For u 2 A, put Similrly, for u 2 A!, put u 1 X = fx 2 A 1 j ux 2 Xg Xu! = fx 2 A + j (xu)! 2 Xg Xu 1 = fx 2 A + j xu 2 Xg Now, n 1-vriety ssocites to every lphet A set A 1 V of recognizle sets of A 1 stisfying the following properties: (1) for ech lphet A, A 1 V contins ;, A + nd A! nd is closed under nite union nd complement, (2) for ech mp ' : A! B +, which denes morphism from ' : A 1! B 1, X 2 B 1 V implies X' 1 2 A 1 V, (3) if X 2 A 1 V, then, for ll u 2 A, u 1 X; Xu! 2 A 1 V nd, for ll u 2 A!, Xu 1 2 A 1 V. Note tht y property (1), if X 2 A 1 V, then X + = X\A + nd X! = X\A! re lso in A 1 V. Furthermore, the clss of lnguges which ssocites to ech lphet A, the set of lnguges of the form X +, where X 2 A 1 V, is +- vriety, clled the +-prt of V. Similrly, the!-prt of V ssocites to ech lphet A, the set of lnguges of the form X!, where X 2 A 1 V. There is lso vriety theorem for 1-vrieties, ut the lgeric counterprt is more involved. Since it will not e used in this rticle, we just refer the interested reder to [4,6] for more informtion. There re severl nturl connections etween lnguges nd!-lnguges. One of the simplest one is to consider nite deterministic utomton s Buchi utomton. In terms of lnguges, this denes, for given lnguge L of A +, the set! L of!-words of A! tht hve n innite numer of prexes in L. It is well-known tht n!-lnguge is of the form! L for some lnguge L if nd only if it is ccepted y deterministic Buchi utomton. For this reson, we will use the term deterministic to designte the!-lnguges of this form. McNughton hs shown tht ny recognizle!-lnguge is oolen comintion of deterministic recognizle!-lnguges. In view of possile extension of McNugthon's theorem to vrieties, Pecuchet considered, for given +-vriety, the clss! V dened s follows: for ech lphet A, A!! V is the set of ll oolen comintions of!-lnguges of the form! L, where L 2 A + V. Wilke [6] sked whether such clsses! V form the!-prt of n 1-vriety. As it ws shown y Perrin [3], the nswer is positive if V is closed
under product, tht is, if, for ech lphet A, L; L 0 2 A + V implies LL 0 2 V. The nswer is lso positive for ll the vrieties studied y Pecuchet [1,2]. The im of this pper is to provide negtive nswer to the question rised y Wilke. Let BA 2 e the ve element Brndt periodic semigroup, tht is, the semigroup with zero presented on f; g y the reltions =, = nd 2 = 2 = 0. This semigroup cn lso e dened s the syntctic semigroup of the lnguge () + over the two-letter lphet f; g, or s the semigroup of prtil functions given in the following tle 3 2 1 2 1 2 1 2 Alterntively, BA 2 is the semigroup of two-y-two mtrices 1 0 0 1 0 0 0 0 0 0 ; ; ; ; 0 0 0 0 1 0 0 1 0 0 under the usul mtrix multipliction. Let V e the vriety of nite semigroups generted y BA 2. It ws shown in [5] tht V is dened y the identities xyx = xyxyx x 2 y 2 = y 2 x 2 x 2 = x 3 (0:1) Let V e the +-vriety corresponding to V. The im of this pper is to prove the following negtive result. Theorem The clss! V is not closed under inverse morphisms. In prticulr, this gives negtive nswer to the question proposed y Wilke. Corollry The clss! V is not the!-prt of n 1-vriety. Proof. Let A = f; g, B = f; ; cg nd ' : A +! B + e the morphism dened y ' = c nd ' =. Let L = f; cg. A simple computtion shows tht the syntctic semigroup of L is BA 2 nd thus L 2 B + V. Let X =! L ' 1. It is esily veried tht X is the set of!-words over A contining n innite numer of 's, tht is X =! A. We clim tht X is not oolen comintion of!-lnguges of the form K!, with K 2 A + V.
4 Let us rst descrie the lnguges of A + V. Every semigroup of V generted y A is quotient of the reltively free semigroup F A V, tht is, the semigroup presented on A y the reltions (1) xyx = xyxyx (2) x 2 y 2 = y 2 x 2 (3) x 2 = x 3 for ll x 2 A + It is esy to derive from (1), (2), (3) the identities (4) xyxzx = xzxyx (5) x 2 y 2 = x 2 y 2 x = x 2 y 3 (6) x 2 yx = x 2 y 2 Indeed, consider the following derivtions, where the identity used t ech step is indicted ove the equlity symol. xyxzx (1) = xyxyxzxzx (2) = xzxzxyxyx (1) = xzxyx x 2 y 2 (2) = y 2 x 2 (3) = y 2 x 3 (2) = x 2 y 2 x x 2 y 2 (3) = x 2 y 3 x 2 yx (1) = xxyxyx (4) = xyxxyx (1) = xyxxyxyx (2) = xyyxyxxx (3) = xyyxyxx (4) = xyxyyxx (2) = xyxxxyy (3) = xyxxyy (2) = xyyyxx (3) = xyyxx (2) = xxxyy (3) = xxyy Using these identities, one gets 17 element semigroup whose right representtion is shown in the grph elow. The edges of this grph re of the form s! s for s 2 F A V nd 2 A, ut the rrows ending in 0 re omitted. Thus = = = = = = = = = = = = 0. 0
For instnce, = since there is n rrow of lel from to. It is not dicult to see from this digrm tht the lnguges recognized y F A V (tht is, the lnguges of A + V) re unions of lnguges of one of the following ctegories: (1) +, +, +, +, () +, () +, () +, () + (2) nite lnguge (3) R = A ( 2 + 2 2 + 2 + 2 + 2 2 + 2 + 3 )A Now, if F is nite, F! = ;. Therefore, every!-lnguge of the form K!, where K 2 A + V, cn e written s union of!-lnguges of the form!,!,!,!, ()!, ()! or! R. But now, if Z is one of these!-lnguges, one hs ()! 2 Z if nd only if! 2 Z. Therefore this property lso holds if Z is oolen comintion of these!-lnguges. Now, since ()! 2 X ut! =2 X, X cnnot e expressed s oolen comintion of!-lnguges of the form K!, where K 2 A + V. Thus X =2 A!! V. 5 Acknowledgements The uthor would like to thnk the nonymous referee for pointing out n error in the computtion of F A V in the rst version of this rticle. References [1] J. P. Pecuchet, Vrietes de semigroupes et mots innis, in B. Monien nd G. Vidl-Nquet eds., STACS 86, Lecture Notes in Comput. Sci. 210, Springer, (1986), 180{191. [2] J. P. Pecuchet, Etude syntxique des prties reconnissles de mots innis, in Proc. 13th ICALP, Kott ed.) Lecture Notes in Comput. Sci. 226, (1986), 294{303. [3] D. Perrin, Vrietes de semigroupes et mots innis, C.R. Acd. Sci. Pris 295, (1982), 595{598. [4] D. Perrin nd J.-E. Pin, Semigroups nd utomt on innite words, Semigroups, Forml Lnguges nd Groups, J. Fountin (ed.), Kluwer Acdemic Pulishers (1995), 49{72. [5] A. N. Trhtmn, Identities of ve-element 0-simple semigroup, Semigroup Forum 48, (1994), 385{387. [6] T. Wilke, An lgeric theory for regulr lnguges of nite nd innite words, Alger nd Computtion 3, (1993), 447{489.