Some conditions from a finite regular semigroup to a finite inverse semigroup

TL;DR

This paper investigates conditions under which a finite regular semigroup can be characterized as a finite inverse semigroup, extending classical representation results from inverse semigroup theory.

Contribution

It provides new sufficient conditions that allow one to identify finite inverse semigroups within finite regular semigroups, complementing Munn's representation theory.

Findings

Identifies specific conditions that guarantee a finite regular semigroup is inverse.

Extends classical representation results to finite semigroups.

Provides criteria useful for algebraic classification of semigroups.

Abstract

In 1978, Munn proved that a bounded complex representation of an inverse semigroup is semiunitary and completely reducible. We consider the converse question in the finite case. We provide some sufficient conditions from a finite regular semigroup to a finite inverse semigroup.