Reducibility of pointlike problems

We show that the pointlike and the idempotent pointlike problems are reducible with respect to natural signatures in the following cases: the pseudovariety of all fi nite semigroups in which the order of every subgroup is a product of elements of a fi xed set of primes; the pseudovariety of all fi nite semigroups in which every regular J-class is the product of a rectangular band by a group from a fixed pseudovariety of groups that is reducible for the pointlike problem, respectively graph reducible. Allowing only trivial groups, we obtain omega-reducibility of the pointlike and idempotent pointlike problems, respectively for the pseudovarieties of all finite aperiodic semigroups (A) and of all finite semigroups in which all regular elements are idempotents (DA).ANR 2010 BLAN 0202 01 FRE

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

Separating Regular Languages by Piecewise Testable and Unambiguous Languages

Abstract. Separation is a classical problem asking whether, given two sets belonging to some class, it is possible to separate them by a set from another class. We discuss the separation problem for regular languages. We give a Ptime algorithm to check whether two given regular languages are separable by a piecewise testable language, that is, whether a BΣ1(<) sentence can witness that the languages are disjoint. The proof refines an algebraic argument from Almeida and the third author. When separation is possible, we also express a separator by saturating one of the original languages by a suitable congruence. Following the same line, we show that one can as well decide whether two regular languages can be separated by an unambiguous language, albeit with a higher complexity.

Ordered monoids and J-trivial monoids

In this paper we give a new proof of the following result of Straubing and Thérien: every J -trivial monoid is a quotient of an ordered monoid satisfying the identity x <= 1

Holonomy Embedding for Arbitrary Stable Semigroups

Original article can be found at: http://ejournals.wspc.com.sg/journals/ijac/mkt/archive.shtml Copyright World Scientific Publishing Company. DOI: 10.1142/S0218196702001206 [Full text of this article is not available in the UHRA]We show how the Rhodes expansion Ŝ of any stable semigroup S embeds into the cascade integral (a natural generalization of the wreath product) of permutation-reset transformation semigroups with zero adjoined. The permutation groups involved are exactly the Schützenberger groups of the -classes of S. Since S ←← Ŝ is an aperiodic map via which all subgroups of S lift to Ŝ, this results in a strong Krohn–Rhodes–Zeiger decomposition for the entire class of stable semigroups. This class includes all semigroups that are finite, torsion, finite -above, compact Hausdorff, or relatively free profinite, as well as many other semigroups. Even if S is not stable, one can expand it using Henckell's expansion and then apply our embedding. This gives a simplified proof of the Holonomy Embedding theorem for all semigroups.Peer reviewe