Profinite Identities for Finite Semigroups Whose Subgroups Belong to a Given Pseudovariety

Free Profinite Semigroups Over Some Classes of Semigroups Locally in D(G)

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...

Decidability and Tameness in the Theory of Finite Semigroups

On Semigroups Whose Idempotent-Generated Subsemigroup is Aperiodic

In this paper, all semigroups are assumed to be finite unless otherwise stated. If H is a pseudovariety of groups (that is, a class of groups closed under formation of finite direct products, subgroups and quotients groups), then it is natural to define a group to be Hsolvable if it has a subnormal series each of whose quotients belongs to H. For instance a group is solvable in the classical se...

Irreducibility of certain pseudovarieties 1

We prove that the pseudovarieties of all finite semigroups, and of all aperiodic fin1 e sernigroups are irreducible for join, for semidirect product and for hlal 'ce~. product. In particular, these pseudovarieties do not admit maximal proper suhpseudovariet~es. More generally, analogous results are proved for the pseudovar~ety of all finite serrigroups all of whose subgroups are in a fixed pseu...

Factoriality and the Pin-Reutenauer procedure

We consider implicit signatures over finite semigroups determined by sets of pseudonatural numbers. We prove that, under relatively simple hypotheses on a pseudovariety V of semigroups, the finitely generated free algebra for the largest such signature is closed under taking factors within the free pro-V semigroup on the same set of generators. Furthermore, we show that the natural analogue of ...