The Booleanization of an inverse semigroup

Mark V. Lawson

doi:10.1007/s00233-019-10071-8

The Booleanization of an inverse semigroup

Mark V. Lawson

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Abstract

We prove that the forgetful functor from the category of Boolean inverse semigroups to the category of inverse semigroups with zero has a left adjoint. This left adjoint is what we term the ‘Booleanization’. We establish the exact theoretical connection between the Booleanization of an inverse semigroup and Paterson’s universal groupoid of the inverse semigroup and we explicitly compute the concrete Booleanization of the polycyclic inverse monoid P_n and demonstrate its affiliation with the Cuntz–Toeplitz algebra.

Original language	English
Pages (from-to)	283–314
Number of pages	32
Journal	Semigroup Forum
Volume	100
Early online date	19 Nov 2019
DOIs	https://doi.org/10.1007/s00233-019-10071-8
Publication status	Published - Feb 2020

Keywords

Boolean algebras
Inverse semigroups
Stone duality

ASJC Scopus subject areas

Algebra and Number Theory

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This is a post-peer-review, pre-copyedit version of an article published in Semigroup Forum. The final authenticated version is available online at: https://doi.org/10.1007/s00233-019-10071-8
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AB - We prove that the forgetful functor from the category of Boolean inverse semigroups to the category of inverse semigroups with zero has a left adjoint. This left adjoint is what we term the ‘Booleanization’. We establish the exact theoretical connection between the Booleanization of an inverse semigroup and Paterson’s universal groupoid of the inverse semigroup and we explicitly compute the concrete Booleanization of the polycyclic inverse monoid Pn and demonstrate its affiliation with the Cuntz–Toeplitz algebra.

KW - Boolean algebras

KW - Inverse semigroups

KW - Stone duality

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JO - Semigroup Forum

JF - Semigroup Forum

ER -

The Booleanization of an inverse semigroup

Abstract

Keywords

ASJC Scopus subject areas

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