Finite regular semigroups with permutations that map elements to inverses

TL;DR

This paper explores permutation matchings in finite regular semigroups, providing partial solutions to open questions and introducing a new combinatorial problem related to elements and their inverses.

Contribution

It offers a comprehensive account of permutation matchings, including partial solutions to open problems and a novel combinatorial problem in the context of finite regular semigroups.

Findings

Partial solutions to open questions on permutation matchings

Introduction of a new combinatorial problem related to inverses

Insights into the structure of permutation matchings in semigroups

Abstract

We give an account on what is known on the subject of permutation matchings, which are bijections of a finite regular semigroup that map each element to one of its inverses. This includes partial solutions to some open questions, including a related novel combinatorial problem.