Transformation Representations of Sandwich Semigroups

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 ⋆_a defined by x⋆_ay = xay for x, y ∈ 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.

KEYWORDS:

• transformation semigroups
• variant semigroups
• sandwich semigroups
• transformation representations
• embeddings

2010 AMS SUBJECT CLASSIFICATION:

• 20M20, 20M30, 20M15, 20M10