Four Notions of Conjugacy for Abstract Semigroups

TL;DR

This paper explores four different notions of conjugacy in semigroups, analyzing their relationships and differences across various mathematical contexts, and discusses open problems in the field.

Contribution

It introduces and compares four notions of conjugacy in semigroups, providing a unified, general framework and highlighting their interconnections and distinctions.

Findings

Four notions of conjugacy are systematically studied.

Relationships and differences among these notions are clarified.

The paper presents numerous open problems for future research.

Abstract

The action of any group on itself by conjugation and the corresponding conjugacy relation play an important role in group theory. There have been many attempts to find notions of conjugacy in semigroups that would be useful in special classes of semigroups occurring in various areas of mathematics, such as semigroups of matrices, operator and topological semigroups, free semigroups, transition monoids for automata, semigroups given by presentations with prescribed properties, monoids of graph endomorphisms, etc. In this paper we study four notions of conjugacy for semigroups, their interconnections, similarities and dissimilarities. They appeared originally in various different settings (automata, representation theory, presentations or transformation semigroups). Here we study them in maximum generality. The paper ends with a large list of open problems.