Closures of regular languages for profinite topologies

Abstract

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 $$\mathsf{A}$$ 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 $$\mathsf{A}$$ , 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 $$\mathsf{V}$$ 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 $$\mathsf{V}$$ . In the cases of $$\mathsf{A}$$ and of the pseudovariety $$\mathsf{DA}$$ of semigroups in which all regular elements are idempotents, this is a new result.