Quotients of the Booleanization of an inverse semigroup

TL;DR

This paper introduces a new class of representations for inverse semigroups called $X$-to-join, constructs their universal Booleanization, and applies this to model boundary quotients of certain $C^*$-algebras using groupoids.

Contribution

It defines $X$-to-join representations, constructs the universal $X$-to-join Booleanization, and connects these to groupoid models for boundary quotients of $C^*$-algebras.

Findings

$X$-to-join Booleanization is a weakly meet-preserving quotient of ${ m B}(S)$

All such quotients arise from $X$-to-join representations

Provides groupoid models for boundary quotients of Zappa-Szép product semigroup $C^*$-algebras

Abstract

We introduce $X$-to-join representations of inverse semigroups which are a relaxation of the notion of a cover-to-join representation. We construct the universal $X$-to-join Booleanization of an inverse semigroup $S$ as a weakly meet-preserving quotient of the universal Booleanization ${\mathrm B}(S)$ and show that all such quotients of ${\mathrm B}(S)$ arise via $X$-to-join representaions. As an application, we provide groupoid models for the intermediate boundary quotients of the $C^*$-algebra of a Zappa-Sz\'ep product right LCM semigroup by Brownlowe, Ramagge, Robertson and Whittaker.