# The algebra of rewriting for presentations of inverse monoids

## TL;DR

This paper develops a formal algebraic framework using groupoids to analyze rewriting systems for inverse monoid presentations, introducing pseudoregular groupoids and a novel approach to relation modules.

## Contribution

It introduces a new formalism based on groupoids and the Squier complex for inverse monoids, including the concept of pseudoregular groupoids and a method to derive relation modules.

## Key findings

- Defined pseudoregular groupoids as fundamental groupoids of the Squier complex.
- Provided a free presentation of the relation module using properties of idempotent separating maps.
- Constructed the module of identities via properties of pseudoregular groupoids.

## Abstract

We describe a formalism, using groupoids, for the study of rewriting for presentations of inverse monoids, that is based on the Squier complex construction for monoid presentations. We introduce the class of pseudoregular groupoids, an example of which now arises as the fundamental groupoid of our version of the Squier complex. A further key ingredient is the factorisation of the presentation map from a free inverse monoid as the composition of an idempotent pure map and an idempotent separating map. The relation module of a presentation is then defined as the abelianised kernel of this idempotent separating map. We then use the properties of idempotent separating maps to derive a free presentation of the relation module. The construction of its kernel - the module of identities - uses further facts about pseudoregular groupoids.