Locally involutive semigroups

TL;DR

This paper introduces locally involutive semigroups, embeds them into ordered groupoids, and explores their connections to inverse semigroups, extending classical correspondences and characterizing when certain semigroups are involutive.

Contribution

It extends the classical ESN-correspondence to locally involutive semigroups and characterizes when left involutive semigroups are involutive, providing new embeddings and equivalences.

Findings

Embedding of locally involutive semigroups into ordered groupoids.

Extension of ESN-correspondence to quasi-involutive semigroups.

Characterization of when a left involutive semigroup étale over S is involutive.

Abstract

We introduce locally involutive semigroups and embed them into the category of ordered groupoids. This embedding restricts to a correspondence between quasi-involutive semigroups and ordered groupoids with mediator, extending the classical ESN-correspondence between inverse semigroups and inductive groupoids. An important subcategory of locally involutive semigroups is formed by left involutive semigroups because the classifying topos of an inverse semigroup S is equivalent to the category of left involutive semigroups \'etale over S [4]. We recover this equivalence from a general adjointness and use the latter to determine when a left involutive semigroup \'etale over S is actually an involutive semigroup. Any left involutive semigroup \'etale over S embeds into an involutive S-algebra as we call it. The underlying semigroup of this algebra is involutive.