Homogeneous completely simple semigroups

TL;DR

This paper classifies homogeneous completely simple semigroups, extending the understanding of their structure and including finite regular cases, building on prior work on homogeneous finite groups.

Contribution

It provides a complete classification of homogeneous completely simple semigroups, including finite regular cases, generalizing Cherlin's results on finite groups.

Findings

Complete classification of homogeneous completely simple semigroups

Finite regular homogeneous semigroups characterized

Extension of Cherlin's work on finite groups

Abstract

A semigroup is completely simple if it has no proper ideals and contains a primitive idempotent. We say that a completely simple semigroup $S$ is a homogeneous completely simple semigroup if any isomorphism between finitely generated completely simple subsemigroups of $S$ extends to an automorphism of $S$. Motivated by the study of homogeneous completely regular semigroups, we obtain a complete classification of homogeneous completely simple semigroups, modulo the group case. As a consequence, all finite regular homogeneous semigroups are described, thus extending the work of Cherlin on homogeneous finite groups.