An algebraic characterization of ample type I groupoids

TL;DR

This paper provides algebraic characterizations of type I and CCR properties for ample Hausdorff groupoids and inverse semigroups, linking these properties to subquotients of their Boolean inverse semigroups.

Contribution

It introduces new algebraic criteria for classifying groupoids and inverse semigroups based on their Boolean inverse semigroup structures.

Findings

Algebraic characterizations of type I and CCR properties for groupoids.

Connections between properties of groupoids and subquotients of their Boolean inverse semigroups.

Extension of algebraic characterizations to inverse semigroups via booleanizations.

Abstract

We give algebraic characterizations of the type I and CCR properties for locally compact second countable, ample Hausdorff groupoids in terms of subquotients of their Boolean inverse semigroups of compact open bisections. It yields in turn algebraic characterizations of both properties for inverse semigroups in terms of subquotients of their booleanizations.