A Presentation for the Dual Symmetric Inverse Monoid

TL;DR

This paper introduces a presentation for the dual symmetric inverse monoid, providing new criteria for inverse monoids generated by completely regular elements, advancing algebraic understanding of these structures.

Contribution

It offers a monoid presentation for the dual symmetric inverse monoid and establishes criteria for when a monoid generated by completely regular elements is inverse.

Findings

Provided a monoid presentation for $mathscr{I}_n^*$

Established criteria for inverse monoids generated by completely regular elements

Enhanced algebraic understanding of inverse monoids

Abstract

The dual symmetric inverse monoid $\mathscr{I}_n^*$ is the inverse monoid of all isomorphisms between quotients of an $n$-set. We give a monoid presentation of $\mathscr{I}_n^*$ and, along the way, establish criteria for a monoid to be inverse when it is generated by completely regular elements.