Equidivisible pseudovarieties of semigroups

TL;DR

This paper characterizes equidivisible pseudovarieties of semigroups, identifying their algebraic properties and closure conditions, and provides a complete classification of such structures.

Contribution

It offers a complete characterization of equidivisible pseudovarieties of semigroups, linking them to Mal'cev products and Karnofsky-Rhodes expansion.

Findings

Pseudovarieties of completely simple semigroups are equidivisible.

Equidivisible pseudovarieties are closed under Mal'cev product on the left with locally trivial semigroups.

They are also characterized by closure under two-sided Karnofsky-Rhodes expansion.

Abstract

We give a complete characterization of pseudovarieties of semigroups whose finitely generated relatively free profinite semigroups are equidivisible. Besides the pseudovarieties of completely simple semigroups, they are precisely the pseudovarieties that are closed under Mal'cev product on the left by the pseudovariety of locally trivial semigroups. A further characterization which turns out to be instrumental is as the non-completely simple pseudovarieties that are closed under two-sided Karnofsky-Rhodes expansion.