Transformation representations of sandwich semigroups

TL;DR

This paper demonstrates that for finite full transformation semigroups, local subsemigroups and variants are structurally equivalent, revealing a deep connection between these two constructions through computational experiments.

Contribution

It establishes an isomorphism equivalence between local subsemigroups and variants of finite full transformation semigroups, a novel insight in semigroup theory.

Findings

Local subsemigroups and variants are isomorphic for finite full transformation semigroups.

The result was discovered through computational experiments using GAP.

This reveals a fundamental structural similarity in these semigroup classes.

Abstract

Let $a$ be an element of a semigroup $S$. The local subsemigroup of $S$ with respect to $a$ is the subsemigroup $aSa$ of $S$. The variant of $S$ with respect to $a$ is the semigroup with underlying set $S$ and operation $\star_a$ defined by $x\star_ay=xay$ for $x,y\in S$. We show that the following classes contain precisely the same semigroups, up to isomorphism: all local subsemigroups of all finite full transformation semigroups; and all variants of all finite full transformation semigroups. This result was discovered as a result of some experiments (and accidents) when working with the Semigroups package for GAP.