Transformation representations of sandwich semigroups
James East

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 be an element of a semigroup . The local subsemigroup of with respect to is the subsemigroup of . The variant of with respect to is the semigroup with underlying set and operation defined by for . 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.
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