Filtering germs: Groupoids associated to inverse semigroups

TL;DR

This paper explores the relationships between different groupoids associated with inverse semigroups, establishing isomorphisms among them and analyzing their structures, especially focusing on filters, tight filters, and ultrafilters.

Contribution

It demonstrates that the groupoid of filters is isomorphic to the groupoid of germs from the inverse semigroup's action, clarifying their structural connections.

Findings

The groupoid of filters is isomorphic to the groupoid of germs.

The isomorphism restricts to tight filters and ultrafilters.

Provides a unified framework for understanding inverse semigroup groupoids.

Abstract

We investigate various groupoids associated to an arbitrary inverse semigroup with zero. We show that the groupoid of filters with respect to the natural partial order is isomorphic to the groupoid of germs arising from the standard action of the inverse semigroup on the space of idempotent filters. We also investigate the restriction of this isomorphism to the groupoid of tight filters and to the groupoid of ultrafilters.