Pointlike reducibility of pseudovarieties of the form V*D

In this paper, we investigate the reducibility property of semidirect products of the form V *D relatively to (pointlike) systems of equations of the form x1 =...= xn, where D denotes the pseudovariety of definite semigroups. We establish a connection between pointlike reducibility of V*D and the pointlike reducibility of the pseudovariety V. In particular, for the canonical signature consisting of the multiplication and the (omega-1)-power, we show that V*D is pointlike-reducible when V is pointlike-reducible.European Regional Development Fund, through the programme COMPETEThis work was supported by the European Regional Development Fund, through the programme COMPETE, and by the Portuguese Government through FCT – Fundação para a Ciência e a Tecnologia, under the project PEst-C/MAT/UI0013/2014

Pointlike sets with respect to R and J

We present an algorithm to compute the pointlike subsets of a finite semigroup with respect to the pseudovariety R of all finite R-trivial semigroups. The algorithm is inspired by Henckell’s algorithm for computing the pointlike subsets with respect to the pseudovariety of all finite aperiodic semigroups. We also give an algorithm to compute J-pointlike sets, where J denotes the pseudovariety of all finite J-trivial semigroups. We finally show that, in contrast with the situation for R, the natural adaptation of Henckell’s algorithm to J computes pointlike sets, but not all of them.Pessoa French-Portuguese project Egide- Grices 11113YMFundação para a Ciência e a Tecnologia (FCT

Complete reducibility of pseudovarieties

The notion of reducibility for a pseudovariety has been introduced as an abstract property which may be used to prove decidability results for various pseudovariety constructions. This paper is a survey of recent results establishing this and the stronger property of complete reducibility for specific pseudovarieties.FCT through the Centro de Matemática da Universidade do Minho and Centro de Matemática da Universidade do Port

Tameness of pseudovariety joins involving R

2000 Mathematics Subject Classification: 20M07 (primary); 20M05, 20M35, 68Q70 (secondary).In this paper, we establish several decidability results for pseudovariety joins of the form VvW, where V is a subpseudovariety of J or the pseudovariety R. Here, J (resp. R) denotes the pseudovariety of all J-trivial (resp. R-trivial) semigroups. In particular, we show that the pseudovariety VvW is (completely) kappa-tame when V is a subpseudovariety of J with decidable kappa-word problem and W is (completely) kappa-tame. Moreover, if W is a kappa-tame pseudovariety which satisfies the pseudoidentity x_1...x_ry^{\omega+1}zt^\omega = x_1... x_ryzt^\omega, then we prove that RvW is also kappa-tame. In particular the joins RvAb, RvG, RvOCR, and RvCR are decidable.União Europeia (UE). Fundo Europeu de Desenvolvimento Regional (FEDER) - POCTI/32817/MAT/2000.International Association for the Promotion of Co-operation with Scientists from the New Independent States (NIS) of the Former Soviet Union (INTAS) - project 99-1224.Fundação para a Ciência e a Tecnologia (FCT)

Recognizing pro-R closures of regular languages

Given a regular language L, we effectively construct a unary semigroup that recognizes the topological closure of L in the free unary semigroup relative to the variety of unary semigroups generated by the pseudovariety R of all finite R-trivial semigroups. In particular, we obtain a new effective solution of the separation problem of regular languages by R-languages

Closures of regular languages for profinite topologies

The Pin-Reutenauer algorithm gives a method, that can be viewed as a descriptive procedure, to compute the closure in the free group of a regular language with respect to the Hall topology. A similar descriptive procedure is shown to hold for the pseudovariety A of aperiodic semigroups, where the closure is taken in the free aperiodic omega-semigroup. It is inherited by a subpseudovariety of a given pseudovariety if both of them enjoy the property of being full. The pseudovariety A, as well as some of its subpseudovarieties are shown to be full. The interest in such descriptions stems from the fact that, for each of the main pseudovarieties V in our examples, the closures of two regular languages are disjoint if and only if the languages can be separated by a language whose syntactic semigroup lies in V. In the cases of A and of the pseudovariety DA of semigroups in which all regular elements are idempotents, this is a new result.PESSOA French-Portuguese project Egide-Grices 11113YM, "Automata, profinite semigroups and symbolic dynamics".FCT -- Fundação para a Ciência e a Tecnologia, respectively under the projects PEst-C/MAT/UI0144/2011 and PEst-C/MAT/UI0013/2011.ANR 2010 BLAN 0202 01 FREC.AutoMathA programme of the European Science Foundation.FCT and the project PTDC/MAT/65481/2006 which was partly funded by the European Community Fund FEDER