The transitivity of primary conjugacy in a class of semigroups

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.