Variants of epigroups and primary conjugacy

TL;DR

This paper investigates the transitivity of primary conjugacy in variants of epigroups, confirming that it holds across all variants within a specific variety, thus extending previous results in semigroup theory.

Contribution

The paper proves that primary conjugacy is transitive in all variants of epigroups within a certain variety, answering an open question in the field.

Findings

Transitivity of primary conjugacy holds in all variants of epigroups in the studied variety.

Confirmed that the property extends beyond completely regular semigroups and their variants.

Provides a broader understanding of conjugacy relations in semigroup variants.

Abstract

In a semigroup $S$ with fixed $c\in S$, one can construct a new semigroup $(S,\cdot_c)$ called a \emph{variant} by defining $x\cdot_c y:=xcy$. Elements $a,b\in S$ are \emph{primarily conjugate} if there exist $x,y\in S^1$ such that $a=xy, b=yx$. This coincides with the usual conjugacy in groups, but is not transitive in general semigroups. Ara\'{u}jo \emph{et al.} proved that transitivity holds in a variety $\mathcal{W}$ of epigroups containing all completely regular semigroups and their variants, and asked if transitivity holds for all variants of semigroups in $\mathcal{W}$. We answer this affirmatively as part of a study of varieties and variants of epigroups.