The Inverse Monoid of Partial Inner Automorphisms of a Semigroup

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.

This paper was originally section 5 of arXiv:2301.04252 . We spun it off into its own paper at the suggestion of the referee. If the arXiv system flags it for substantial text overlap, that's why. arXiv admin note: substantial text overlap with arXiv:2301.04252