Conjugacy in inverse semigroups

TL;DR

This paper explores the concept of conjugacy in inverse semigroups, extending the group-theoretic notion to various classes of inverse semigroups and analyzing their properties.

Contribution

It introduces and studies the conjugacy relation in inverse semigroups across multiple classes, providing a comprehensive analysis of its structure and implications.

Findings

Conjugacy in inverse semigroups generalizes group conjugacy.

Characterization of conjugacy in symmetric inverse semigroups.

Analysis of conjugacy in specific classes like Clifford and bicyclic semigroups.

Abstract

In a group $G$, elements $a$ and $b$ are conjugate if there exists $g\in G$ such that $g^{-1} ag=b$. This conjugacy relation, which plays an important role in group theory, can be extended in a natural way to inverse semigroups: for elements $a$ and $b$ in an inverse semigroup $S$, $a$ is conjugate to $b$, which we will write as $a\sim_{\mathrm{i}} b$, if there exists $g\in S^1$ such that $g^{-1} ag=b$ and $gbg^{-1} =a$. The purpose of this paper is to study the conjugacy $\sim_{\mathrm{i}}$ in several classes of inverse semigroups: symmetric inverse semigroups, free inverse semigroups, McAllister $P$-semigroups, factorizable inverse monoids, Clifford semigroups, the bicyclic monoid and stable inverse semigroups.