Equidivisibility and profinite coproduct

TL;DR

This paper studies how equidivisibility behaves under coproducts in the category of pro-V semigroups, introducing new notions like KR-cover and characterizing equidivisible semigroups with letter super-cancellativity.

Contribution

It introduces the concepts of KR-cover and strong KR-cover, and characterizes equidivisible profinite semigroups with letter super-cancellativity, advancing understanding of coproduct behavior.

Findings

KR-cover is stronger than equidivisibility

Characterization of equidivisible profinite semigroups with letter super-cancellativity

Closure of certain classes under finite V-coproducts when V is closed under two-sided Karnofsky--Rhodes expansion

Abstract

The aim of this work is to investigate the behavior of equidivisibility under coproduct in the category of pro-$\mathsf{V}$ semigroups, where $\mathsf{V}$ 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 $\mathsf{V}$ is closed under two-sided Karnofsky--Rhodes expansion, closure of some classes of equidivisible pro-$\mathsf{V}$ semigroups under(finite) $\mathsf{V}$-coproduct is established.