The Bideterministic Concatenation Product

This paper is devoted to the study of the bideterministic concatenation product, a variant of the concatenation product. We give an algebraic characterization of the varieties of languages closed under this product. More precisely, let V be a variety of monoids, V the corresponding variety of languages and V̂ the smallest variety containing V and the bideterministic products of two languages of V . We give an algebraic description of the variety of monoids b V corresponding to V̂ . For instance, we compute b V when V is one of the following varieties : the variety of idempotent and commutative monoids, the variety of monoids which are semilattices of groups of a given variety of groups, the variety ofR-trivial and idempotent monoids. In particular, we show that the smallest variety of languages closed under bideterministic product and containing the language f1g, corresponds to the variety of J -trivial monoids with commuting idempotents. Similar results were known for the other variants of the concatenation product, but the corresponding algebraic operations on varieties of monoids were based on variants of the semidirect product and of the Malcev product. Here the operation V ! b V makes use of a construction which associates to any nite monoid M an expansion c M , with the following properties: (1) M is a quotient of c M , (2) the morphism : c M ! M induces an isomorphism between the submonoids of c M and of M generated by the regular elements and (3) the inverse image under of an idempotent of M is a 2-nilpotent semigroup. This paper assumes some familiarity with Eilenberg's theory of varieties and especially the notion of syntactic monoid of a recognizable language. References for this theory are [6,7,10]. The main result of this theory states that there exists a one-to-one correspondence between certain families of recognizable sets (the varieties of languages) and certain families of nite semigroups (the varieties of nite semigroups). A fundamental result of Sch utzenberger [13] states that the smallest variety of languages closed under concatenation product corresponds to the variety of aperiodic monoids. Since then, an important part of the existing literature on varieties has been devoted to the study of the concatenation product and its variants. These variants include the weak forms of the concatenation product introduced by Sch utzenberger [14] (the unambiguous product and the left and right deterministic products) and the counter product introduced by Straubing [15]. This paper is devoted to the study of the two-sided version of the deterministic products, called the bideterministic product. The general setting for this type of result can be summarized as follows. Let be a binary operation on languages | in our case the concatenation product or one of its variants ( ) Research on this paper was supported for the rst author by PRC "Math ematiques et Informatique" and for the second author by the NSERC grant no. A4546 et FCAR grant no 89-EQ-2933. 2 The bideterministic concatenation productApril 10, 1993 | and let V be the variety of languages corresponding to a variety of monoids V. Denote by V 0 the smallest variety containing V and closed under . The question is to describe the varieties of monoidsV0 corresponding to V 0. For all the variants of concatenation mentionned previously, the variety V0 is equal to a Malcev product of the form W M V, where W is a certain variety of semigroups [16,8,9,12,17,18]. This variety W is given in the following table: Product type Variety W such that V0 =W M V concatenation aperiodic semigroups unambiguous semigroups S such that eSe = e for each idempotent e 2 S right deterministic semigroups S such that eS = e for each idempotent e 2 S left deterministic semigroups S such that Se = e for each idempotent e 2 S counter semigroups which are locally solvable groups This is no longer true for the bideterministic product: in this case, the variety V0 cannot be written as a Malcev product of some variety with V and a new algebraic operation is required. This new operation relies on a construction of independent interest, which associates to any monoid M a certain expansion c M , with the following properties: M is a quotient of c M and the morphism : c M !M induces an isomorphism from hReg(c M)i, the submonoid of c M generated by the regular elements of c M , onto hReg(M)i. Furthermore,the inverse image under of an idempotent of M is a 2-nilpotent semigroup. Our construction is somewhat reminiscent of the expansion proposed by Birget, Margolis and Rhodes in [4,5], but turns out to be di erent, as we shall see on an example. Now the key result states that a variety of languages is closed under bideterministic product if and only if the corresponding variety of monoids is closed under this expansion. We also give a more precise version of this result. Let V be a variety of languages and let V be the corresponding variety of monoids. Let V̂ be the smallest variety containing V and the bideterministic products of two languages of V. Then the variety of monoids corresponding to V̂ is the variety of monoids generated by the monoids of the form M̂ for some M 2 V. Similar results are known for the other variants of products, but again, there are based on totally di erent algebraic constructions (essentially variants of the semidirect product). We compute b V for various varities V, including the variety of idempotent and commutative monoids, the variety of monoids which are semilattices of groups of a given variety of groups and the variety of R-trivial and idempotent monoids. As a byproduct, we characterize the smallest non trivial variety of languages containing the language f1g and closed under bideterministic product : the corresponding variety of monoids is the variety of J -trivial monoids whose idempotents commute. The bideterministic concatenation productApril 10, 1993 3 1. Some preliminaries. In this section, we recall some basic de nitions or facts about nite semigroups and languages. All semigroups and monoids considered in this paper are either nite or free, although some results could be easily extended to periodic semigroups. Let S be a semigroup. We denote by S1 the semigroup equal to S if S has an identity and to S [ f1g, where 1 is a new identity, otherwise. We denote by E(S) the set of idempotents of S. For each element s of S, the subsemigroup of S generated by s contains a unique idempotent, denoted s!. If P is a subset ofM , hP i denotes the submonoid generated by P . Given s; t 2 S, we say that s is R-below t (denoted s R t) if there exists x 2 S1 such that s = tx. The elements s and t are R-equivalent (denoted s R t) if s R t and t R s. Finally, we denote s <R t if s is R-below t but is not R-equivalent with t. The relations L, L and <L are de ned dually. For instance, s L t if there exists x 2 S1 such that s = xt. Let s be a semigroup and let s be an element of S. An element s of S is called a weak inverse of s if ss s = s. It is an inverse of s if ss s = s and s ss = s. In this case, s is an inverse of s. An element which has an inverse is called regular. We denote by Reg(S) the set of regular elements of a semigroup S. The following propositions state some elementary properties of weak inverses. Proposition 1.1. Let s be a weak inverse of s. Then s s and ss are idempotent and s is an inverse of s ss. Proof. If s is a weak inverse of s, we have ss s = s. This implies in particular s ss s = s s and ss ss = ss and thus s s and ss are idempotent. We also have (s ss) s(s ss) = (s s)(s s)(s s)s = s ss and s(s ss) s = s Thus s is an inverse of s ss. Proposition 1.2. Let s and t be elements of a semigroup S such that s R st (resp. ts L s). Then there exists a weak inverse t of t such that st t = s (resp. tts = s). Proof. Since s R st, there exists an element t0 2 S1 such that stt0 = s. Let ! be an integer such that (tt0)! is idempotent, and set t = t0(tt0)2! 1. Then st t = stt0(tt0)2! 1 = s. Furthermore tt t = t0(tt0)2! 1tt0(tt0)2! 1 = t0(tt0)4! 1 = t0(tt0)2! 1 = t. Thus t is a weak inverse of t. The proof for the L relation is dual. A monoid M divides a monoid N if M is a quotient of a submonoid of N . A variety of nite monoids is a class of nite monoids closed under taking submonoids, quotients and nite direct products. Recall that a relational morphism between monoidsM and N is a relation :M ! N such that: (1) (m )(n ) (mn) for all m;n 2M , (2) (m ) is non-empty for all m 2M , (3) 1 2 1 Equivalently, is a relation whose graph graph( ) = f (m;n) j n 2 m g 4 The bideterministic concatenation productApril 10, 1993 is a submonoid of M N that projects onto M . Let V and W be varieties. The Malcev product of V and W is the variety V M W de ned as follows V M W = f M j There is a relational morphism :M ! N with N 2W and such that e 1 2 V for all idempotents e 2 N g Let A be a nite set, called the alphabet, whose elements are letters. We denote by A the free monoid over A. Elements of A are words. In particular, the empty word, denoted by 1, is the identity of A . A language is a subset of A . Let M be a monoid and L be a language of A . A monoid morphism ' : A ! M recognizes a language L if there exists a subset P ofM such that L = ' 1(P ). The syntactic congruence of L is the equivalence L on A de ned by u L v if and only if, for every x; y 2 A (xuy 2 L() xvy 2 L): The quotient A = L is the syntactic monoid of L and the natural morphism : A !M(L) is called the syntactic morphism: it recognizes L and every surjective morphism ' : A !M that recognizes L can be factorized through it, that is, there is a surjective morphism :M !M(L) such that = '. For technical reasons, it is more appropriate to use a variant of the concatenation product called the marked product. The results stated in the introduction refer to this product. Given a nite alphabet A and a letter a of A, the marked product of two subsets (also called languages) L0 and L1 of the free monoid A is the language L0aL1 = fu 2 A j u = u0au1 for some u0 2 L0 and u1 2 L1g Unambiguous, left and right deterministic products were introduced by Sch utzenberger. A product L = L0aL1 is unambiguous if every word u of L has a unique decomposition of the form u = u0au1 with u0 2 L0 and u1 2 L1. It is left deterministic if every word of L has exactly one pre x in L0a. This means that in order to nd the decomposition u = u0au1 of a word of L, it su ces to read u from left to right: the rst pre x of u in L0a will give u0a, and thus the decomposition. Dually, a product L = L0aL1 is right deterministic if every word of L has exactly one su x in aL1. A product is called bideterministic if it is both deterministic and antideterministic. Sch utzenberger [14] characterized the smallest variety of languages containing the language f1g and closed under unambiguous (resp. deterministic, antideterministic) products. Later on, it was shown in [8,9,12] that the closure of a variety of languages under unambiguous (resp. left deterministic, right deterministic) product correspond to the Malcev product V ! LI M V (resp. V ! K M V, V ! Kr M V), where LI, K and Kr are respectively the varieties of semigroups S such that, for every idempotent e 2 S, eSe = e, (resp. eS = e, Se = e). 2. An expansion. In this section, we give the formal de nition of our new expansion, which is related to certain special factorizations of words. Let M be a monoid, and let ' : A ! M be a surjective (monoid) morphism. A good factorization (with respect to ') is a triple (x0; a; x1) 2 A A A such that '(x0a) <R '(x0) and '(ax1) <L '(x1). A good factorization of a word x 2 A is a good factorization (x0; a; x1) such that x = x0ax1. Two good factorizations (x0; a; x1) and (y0; b; y1) are equivalent if '(x0) = '(y0), '(x1) = '(y1) and a = b. In particular, this implies '(x0ax1) = '(y0by1). Here is a rst useful lemma.