Conjugacy in Abstract Semigroups, Transformation and Diagram Monoids, and Conjugacy Growth

TL;DR

This paper explores conjugacy relations in various semigroups and monoids, classifies conjugacy classes in key transformation and diagram monoids, and analyzes conjugacy growth, providing new insights and open problems.

Contribution

It introduces a novel conjugacy relation $ abla$ with elegant properties, classifies conjugacy classes in important semigroups, and studies conjugacy growth asymptotics.

Findings

Complete classification of $ abla$-classes in $ ext{T}_n$, $ ext{I}_n$, and endomorphism monoids.

Analysis of conjugacy relations in diagram semigroups like partition and Brauer monoids.

Asymptotic estimate of conjugacy growth in polycyclic monoids.

Abstract

We study conjugacy relations on semigroups and monoids, focusing on the relation $a \cfn b$, defined by the existence of $g,h \in S^1$ such that $ag = gb$, $bh = ha$, $hag = b$, and $gbh = a$. This notion emerged as one that yields particularly elegant results. The interplay between $\cfn$ and other standard conjugacy relations is analyzed, and some results on special classes of abstract semigroups are established. We then specialize to the case of transformation semigroups. A complete classification of $\cfn$-classes is obtained for the full transformation monoid $\mathcal{T}_n$, the symmetric inverse monoid $\mathcal{I}_n$, and the endomorphism monoid of $G$-sets, among others. We also investigate the natural conjugacy in diagram semigroups, including the partition monoid, the Brauer monoid, and the partial Brauer monoid. Finally, we investigate the conjugacy growth function in polycyclic monoids and obtain a precise asymptotic estimate. The paper concludes with some open problems.