Coverages on Inverse Semigroups

TL;DR

This paper introduces a new, weaker definition of coverage on inverse semigroups, constructs a universal pseudogroup, and explores the connections to topological groupoids and tight filters, advancing the theoretical framework in this area.

Contribution

It generalizes existing coverage definitions on inverse semigroups, constructs a universal pseudogroup, and links nuclei on pseudogroups to topological groupoid embeddings.

Findings

Existence of a universal pseudogroup for a given coverage.

A method to derive topological groupoid embeddings from nuclei.

Application to Exel's tight filters and tight groupoids.

Abstract

First we give a definition of a coverage on a inverse semigroup that is weaker than the one gave by a Lawson and Lenz and that generalizes the definition of a coverage on a semilattice given by Johnstone. Given such a coverage, we prove that there exists a pseudogroup that is universal in the sense that it transforms cover-to-join idempotent-pure maps into idempotent-pure pseudogroup homomorphisms. Then, we show how to go from a nucleus on a pseudogroup to a topological groupoid embedding of the corresponding groupoids. Finally, we apply the results found to study Exel's notions of tight filters and tight groupoids.