Conjugation in Semigroups

TL;DR

This paper introduces a new, broadly applicable definition of conjugacy in semigroups, compares it with existing notions, and analyzes conjugacy classes in transformation semigroups, especially for infinite sets.

Contribution

It proposes a novel conjugacy concept for all semigroups that avoids trivialization in zero semigroups and provides characterizations and counts of conjugacy classes.

Findings

New conjugacy definition applicable to all semigroups

Comparison with existing conjugacy notions

Counting conjugacy classes in transformation semigroups for infinite sets

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 several attempts to extend the notion of conjugacy to semigroups. In this paper, we present a new definition of conjugacy that can be applied to an arbitrary semigroup and it does not reduce to the universal relation in semigroups with a zero. We compare the new notion of conjugacy with existing definitions, characterize the conjugacy in various semigroups of transformations on a set, and count the number of conjugacy classes in these semigroups when the set is infinite.