Variants of epigroups and primary conjugacy

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. Araújo \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.

7 pages; v2: stylistic changes suggested by the referee

References in corpus (2)

• Four Notions of Conjugacy for Abstract Semigroups
• Semigroups of rectangular matrices under a sandwich operation