The algebra of rewriting for presentations of inverse monoids

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 factorization 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 abelianized 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.