# Permutations of a semigroup that map to inverses

## TL;DR

This paper explores conditions under which elements of a finite regular semigroup can be permuted to map each element to an inverse, using graph theory and Hall's Marriage Lemma, with a focus on the full transformation semigroup.

## Contribution

It reformulates the inverse permutation problem in semigroups using graph theory and proves that the finite full transformation semigroup has this property.

## Key findings

- Finite full transformation semigroup can be permuted to map elements to inverses
- Graph-theoretic reformulation of the inverse permutation problem
- Application of Hall's Marriage Lemma to semigroup structure

## Abstract

We investigate the question as to when the members of a finite regular semigroup may be permuted in such a way that each member is mapped to one of its inverses. In general this is not possible. However we reformulate the problem in terms of a related graph and, using an application of Hall's Marriage Lemma, we show in particular that the finite full transformation semigroup does enjoy this property.