COMPLETE REDUCIBILITY OF THE PSEUDOVARIETY LS1

JOSÉ CARLOS COSTA, CONCEIÇÃO NOGUEIRA
2009 International journal of algebra and computation  
In this paper we prove that the pseudovariety LSl of local semilattices is completely κ-reducible, where κ is the implicit signature consisting of the multiplication and the ω-power. Informally speaking, given a finite equation system with rational constraints, the existence of a solution by pseudowords of the system over LSl implies the existence of a solution by κ-words of the system over LSl satisfying the same constraints. plete tameness; complete reducibility; local semillatice; infinite
more » ... rd. the signature σ, and such algebras are called σ-semigroups. Given an alphabet A and a pseudovariety V, the V-free σ-semigroup over A is denoted by Ω σ A V and its elements are called σ-terms over V. A pseudovariety V is said to be σ-tame if the word problem for σ-terms over V is decidable, and if V is σ-reducible, which means, informally speaking, the following: if a finite graph equation system with rational constraints has a solution in the free profinite semigroup "modulo" V, then it has also a solution given by σ-terms and satisfying the same constraints. More generally, when not only graph equation systems but every finite systems of σ-term equations are considered, V is said to be completely σ-reducible (and completely σ-tame when, furthermore, the word problem for σ-terms over V is decidable). This extension of the notion of reducibility, was defined by Almeida [5] mainly because various kinds of such more general systems appear when different pseudovariety operators are considered. Moreover, unlike reducibility, the complete reducibility property is inherited by the dual pseudovariety. The notion of complete reducibility was introduced independently by Rhodes and Steinberg [21], using the different terminology of inevitable substitutions, in a more general setting, that of pseudovarieties of relational morphisms and not just of semigroups. The implicit signature which is most commonly encountered in the literature is the canonical signature κ = {ab, a ω−1 } consisting of two pseudowords: ab, representing semigroup multiplication, and a ω−1 , the unary pseudoinverse which, evaluated on an element s of a profinite semigroup, takes the value of the unique inverse of se in the maximal subgroup of the closed subsemigroup generated by s, where e denotes the idempotent of that subgroup. There are several examples of reducibility results in the literature but relatively few results of complete reducibility. We recall some of these results. It is known, for instance, that the pseudovariety G of all finite groups is κ-tame [12, 3, 10] but it is not completely κ-tame [16] . Whether G is completely σ-tame for some other signature σ is still an open question. The pseudovariety G p of all finite p-groups, with p prime, is not κ-tame [10] but Almeida has exhibited an infinite implicit signature with respect to which it is tame [4] . The pseudovariety Ab of all finite abelian groups is completely κ-tame [9] . For aperiodic examples, we should mention the pseudovarieties J and R of all finite semigroups in which, respectively, the Green relations J and R are trivial. The complete κ-tameness of J is proved explicitly in [5], but it is also implicit in the proofs of [22] as is the κ-tameness of J ∨ G. The proof of complete κ-tameness of R was obtained recently [7, 8] , while its κ-tameness had already been established by the same authors in [6] , where several joins involving R, namely R ∨ G, were also shown to be κ-tame. Let LSl be the pseudovariety of all finite semigroups which are locally semilattices, that is, semigroups S such that eSe ∈ Sl for all idempotents e ∈ S. Notice that LSl is the pseudovariety associated, via Eilenberg's correspondence, to the variety of locally testable languages, which as one recalls is formed by the languages L whose membership of a given word u in L can be decided by considering the factors of a fixed length n of u and its prefix and suffix of length n − 1. This pseudovariety is already known to be κ-tame [15] . A weaker property, the pointlike subsets of a finite semigroup being decidable with respect to LSl, was first proved in [23] . In this paper, we extend the above mentioned work of the first author and Teixeira [15] by proving the complete κ-tameness of LSl. This work is organized as follows. After a section of preliminaries, where we introduce some notation and review some basic results on semigroups, pseudovarieties and words,
doi:10.1142/s021819670900507x fatcat:yseeuvsfdjfk3jrmi2vzu3rzf4