# Orthodox semigroups and permutation matchings

## TL;DR

This paper characterizes when orthodox semigroups admit permutation matchings to their inverses, showing such permutations can be involutions, with effective conditions for finite cases and applications to class characterization.

## Contribution

It provides necessary and sufficient conditions for the existence of permutation matchings in orthodox semigroups, including finite cases, and characterizes certain semigroup classes via these matchings.

## Key findings

- Permutation matchings exist under specific conditions involving Green's relations.
- Such permutations can always be chosen as involutions.
- Finite orthodox semigroups have an effective criterion for permutation matchings.

## Abstract

We determine when an orthodox semigroup S has a permutation that sends each member of S to one of its inverses and show that if such a permutation exists, it may be taken to be an involution. In the case of a finite orthodox semigroup the condition is an effective one involving Green's relations on the combinatorial images of the principal factors of S. We also characterise some classes of semigroups via their permutation matchings.