The tight groupoid of an inverse semigroup

TL;DR

This paper establishes algebraic conditions on inverse semigroups that determine when their associated tight groupoids possess properties like being Hausdorff, essentially principal, minimal, and contracting, with some conditions being both necessary and sufficient.

Contribution

It provides a set of algebraic criteria linking inverse semigroup properties to topological features of their tight groupoids, clarifying when these properties are equivalent.

Findings

Conditions for the tight groupoid to be Hausdorff

Criteria for the groupoid to be essentially principal

Characterization of minimal and contracting groupoids

Abstract

In this work we present algebraic conditions on an inverse semigroup S (with zero) which imply that its associated tight groupoid G_{tight)(S) is: Hausdorff, essentially principal, minimal and contracting, respectively. In some cases these conditions are in fact necessary and sufficient.