The Booleanization of an inverse semigroup

TL;DR

This paper introduces the Booleanization functor for inverse semigroups, linking it to groupoid theory and C*-algebras, providing explicit computations for key examples.

Contribution

It establishes the existence of a left adjoint functor called Booleanization for inverse semigroups and connects it to Paterson's universal groupoid and C*-algebra representations.

Findings

Booleanization is explicitly computed for the polycyclic inverse monoid P_n

Booleanization relates to Paterson's universal groupoid

Connection established with the Cuntz-Toeplitz algebra

Abstract

We prove that the forgetful functor from the category of Boolean inverse semigroups to inverse semigroups with zero has a left adjoint. This left adjoint is what we term the `Booleanization'. We establish the exact connection between the Booleanization of an inverse semigroup and Paterson's universal groupoid of the inverse semigroup and we explicitly compute the Booleanization of the polycyclic inverse monoid $P_{n}$ and demonstrate its affiliation with the Cuntz-Toeplitz algebra.