Local subsemigroups and variants of some class of semigroups

TL;DR

This paper investigates the structure of local subsemigroups and variants within certain classes of semigroups, specifically focusing on full transformation semigroups and symmetric inverse monoids, revealing structural similarities.

Contribution

It characterizes local subsemigroups and variants of full transformation semigroups and symmetric inverse monoids, showing their structural equivalence in finite cases.

Findings

Local subsemigroups of full transformation semigroups are characterized.

Variants of symmetric inverse monoids are structurally similar to their local subsemigroups.

Finite symmetric inverse monoids have isomorphic sets of local subsemigroups and variants.

Abstract

For an element a in a semigroup S the local subsemigroup of S with respect to a is the subsemigroup aSa of S and the variant of S with respect to a is a semigroup with underlying set S with a sandwich operation xy = xay for all x, y in S. In this paper we discuss the structures of local subsemigroups of full transformation semigroups and symmetric inverse monoids. It is also shown that the set of all local subsemigroups of finite symmetric inverse monoids and the set of all variants of all finite symmetric inverse monoids are same upto isomorphism.