Equidivisibility and profinite coproduct

Abstract

The aim of this work is to investigate the behavior of equidivisibility under coproduct in the category of pro-V semigroups, where V is a  pseudovariety of finite semigroups. Exploring the relationship with the two-sided Karnofsky–Rhodes expansion, the notions of KR-cover  and strong KR-cover for profinite semigroups are introduced. The former is stronger than equidivisibility and the latter provides a  characterization of equidivisible profinite semigroups with an extra mild condition, so-called letter super-cancellativity. Furthermore,  under the assumption that V is closed under two-sided Karnofsky–Rhodes expansion, closure of some classes of equidivisible pro-V  semigroups under (finite) V-coproduct is established.