Variants of epigroups and primary conjugacy

Abstract

In a semigroup S with fixed $c\in S,$ one can construct a new semigroup $(S,{·}_c)$ called a variant by defining $x{·}_{c}y:=\mathit{xcy}.$ Elements $a,b\in S$ are 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 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.

Keywords:

• Conjugacy
• semigroup
• variants

1991 MATHEMATICS SUBJECT CLASSIFICATION:

• 20M07

Acknowledgement

This article is part of the first author’s dissertation for the PhD Program in Computational Algebra at Universidade Aberta in Portugal. Most of the proofs were obtained with the assistance of the automated theorem prover Prover9 developed by McCune [30].

Additional information

Funding

Partially supported by Simons Foundation Collaboration Grant 359872 and by Fundação para a Ciência e a Tecnologia (Portuguese Foundation for Science and Technology) grant PTDC/MAT-PUR/31174/2017.