Let H be a pseudovariety of groups and DRH be the pseudovariety containing all finite semigroups whose regular R-classes belong to H. We study the relationship between reducibility of H and of DRH with respect to several particular classes of systems of equations. The classes of systems considered (of pointlike, idempotent pointlike and graph equations) are known to play a role in decidability ...

The theory of free profinite semigroups, which received its major impetus with the publication of Reiterman’s paper [17] in the early eighties, has proven to be an important tool in the study of pseudovarieties of semigroups and on the varieties of recognizable languages associated with them (via Eilenberg’s Theorem on varieties [13]). The importance of Reiterman’s theorem was immediately under...

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 ω-semigroup. It is inherited by a subpseudovariety of a give...

The notion of Abelian kernel of a finite monoid extends the notion of derived subgroup of a finite group. In this line, an extension of the notion of solvable group to monoids is quite natural: they are the monoids such that the chain of Abelian kernels ends with the submonoid generated by the idempotents. We prove in this paper that the finite idempotent commuting monoids satisfying this prope...

Radicals for Fitting pseudovarieties of groups are investigated from a profinite viewpoint in order to describe Malcev products on the left by the corresponding local pseudovariety of semigroups.

A class TC(X) of X-graphs is introduced, and an algorithmic property of labellings of nite strongly connected graphs by rational languages is shown to be decidable. This property, named as TC-inevitability, is a language-theoretic analogue of the R-inevitability of labellings of nite strongly connected graphs by nite monoids. As a consequence, the pseudovariety R of all nite R-trivial monoids i...