The transitivity of primary conjugacy in a class of semigroups

TL;DR

This paper investigates the conditions under which primary conjugacy in certain semigroups is transitive, specifically showing transitivity in semigroups where the product of elements satisfies specific algebraic conditions.

Contribution

It establishes that primary conjugacy is transitive in semigroups satisfying the condition that products are either commutative or powers of the product, addressing an open problem.

Findings

Primary conjugacy is transitive in semigroups with specific product conditions.

Semigroups satisfying $xy ext{ in } ext{}\{yx, (xy)^n ext{} ext{ for } n>1$ have transitive primary conjugacy.

Abstract

Elements $a,b$ of a semigroup $S$ are said to be \emph{primarily conjugate} or just \emph{p-conjugate}, if there exist $x,y\in S^1$ such that $a=xy$ and $b=yx$. The p-conjugacy relation generalizes conjugacy in groups, but for general semigroups, it is not transitive. Finding the classes of semigroups in which this notion is transitive is an open problem. The aim of this note is to show that for semigroups satisfying $xy\in\{yx, (xy)^n\}$ for some $n>1$, primary conjugacy is transitive.