The etale groupoid of an inverse semigroup as a groupoid of filters

TL;DR

This paper simplifies the construction of the etale groupoid of an inverse semigroup by using filters, revealing new connections between filters, representations, and partial bijections.

Contribution

It introduces a streamlined filter-based approach to construct the topological groupoid of an inverse semigroup, linking filters to representations via partial bijections.

Findings

The topological groupoid is a groupoid of filters.

Idempotent filters are closed inverse subsemigroups.

Linear representations can be derived from groups in the groupoid.

Abstract

Paterson showed how to construct an etale groupoid from an inverse semigroup using ideas from functional analysis. This construction was later simplified by Lenz. We show that Lenz's construction can itself be further simplified by using filters: the topological groupoid associated with an inverse semigroup is precisely a groupoid of filters. In addition, idempotent filters are closed inverse subsemigroups and so determine transitive representations by means of partial bijections. This connection between filters and representations by partial bijections is exploited to show how linear representations of inverse semigroups can be constructed from the groups occuring in the associated topological groupoid.