The Inverse Monoid of Partial Inner Automorphisms of a Semigroup

TL;DR

This paper introduces an inverse monoid structure for partial inner automorphisms of semigroups, generalizing known automorphism groups and exploring specific cases like groups and transformation monoids.

Contribution

It defines a new inverse monoid associated with semigroups and characterizes it for various classes, extending the understanding of automorphism structures.

Findings

Inverse monoid of partial inner automorphisms generalizes automorphism groups.

For groups, the inverse monoid is isomorphic to the group of inner automorphisms with zero.

Describes the structure for completely simple semigroups, transformation monoids, and endomorphism monoids of finite G-sets.

Abstract

We introduce the inverse monoid of inner partial automorphisms of a semigroup -- a tool that associates to every semigroup an inverse semigroup. When the semigroup is a group, this inverse semigroup is isomorphic to the group of inner automorphisms with a zero adjoined. We then describe this structure for completely simple semigroups, the full transformation monoid, and the endomorphism monoid of a finite $G$-set when $G$ is a finite abelian group. The paper ends with some open problems.