# Homogeneity of Inverse Semigroups

## TL;DR

This paper investigates the conditions under which inverse semigroups are homogeneous, exploring their classifications and extending known results from semilattices and groups to this broader algebraic context.

## Contribution

It establishes when different notions of homogeneity in inverse semigroups are equivalent and classifies certain types of homogeneous inverse semigroups, including periodic commutative ones.

## Key findings

- Homogeneity notions are equivalent under certain conditions.
- Classification of periodic commutative inverse semigroups.
- Extension of classifications from semilattices and groups.

## Abstract

An inverse semigroup $S$ is a semigroup in which every element has a unique inverse in the sense of semigroup theory, that is, if $a \in S$ then there exists a unique $b\in S$ such that $a = aba$ and $b = bab$. We say that an inverse semigroup $S$ is a homogeneous (inverse) semigroup if any isomorphism between finitely generated (inverse) subsemigroups of $S$ extends to an automorphism of $S$. In this paper, we consider both these concepts of homogeneity for inverse semigroups, and show when they are equivalent. We also obtain certain classifications of homogeneous inverse semigroups, in particular periodic commutative inverse semigroups. Our results extend both the classification of homogeneous semilattices and the classification of certain classes of homogeneous groups, in particular the homogeneous abelian groups and homogeneous finite groups.