Representations of relatively free profinite semigroups, irreducibility, and order primitivity

TL;DR

This paper investigates the structure of relatively free profinite semigroups within certain pseudovarieties, demonstrating their faithful action on minimal ideals and establishing join irreducibility results, including for Krohn-Rhodes complexity pseudovarieties.

Contribution

It extends join irreducibility results to broader classes of pseudovarieties and proves the stronger join irreducibility for Krohn-Rhodes complexity pseudovarieties.

Findings

Relatively free profinite semigroups act faithfully on their minimal ideals.

Several pseudovarieties are shown to be join irreducible in the lattice of ordered semigroups.

Proves stronger join irreducibility for Krohn-Rhodes complexity pseudovarieties.

Abstract

We establish that, under certain closure assumptions on a pseudovariety of semigroups, the corresponding relatively free profinite semigroups freely generated by a non-singleton finite set act faithfully on their minimum ideals. As applications, we enlarge the scope of several previous join irreducibility results for pseudovarieties of semigroups, which turn out to be even join irreducible in the lattice of pseudovarieties of ordered semigroups, so that, in particular, they are not generated by proper subpseudovarieties of ordered semigroups. We also prove the stronger form of join irreducibility for the Krohn-Rhodes complexity pseudovarieties, thereby solving a problem proposed by Rhodes and Steinberg.