A dualizing object approach to non-commutative Stone duality

TL;DR

This paper extends the dualizing object approach to Stone duality into the non-commutative realm of skew Boolean algebras, establishing dual adjunctions and embeddings that generalize classical duality theories.

Contribution

It introduces a duality framework for skew Boolean algebras using dualizing objects, expanding non-commutative Stone duality and constructing canonical embeddings.

Findings

Constructed dual adjunctions between Boolean spaces and skew Boolean algebras.

Described Eilenberg-Moore categories of the associated monads.

Provided a non-commutative reflection and embedding of skew Boolean algebras.

Abstract

The aim of the present paper is to extend the dualizing object approach to Stone duality to the non-commutative setting of skew Boolean algebras. This continues the study of non-commutative generalizations of different forms of Stone duality initiated in recent papers by Bauer and Cvetko-Vah, Lawson, Lawson and Lenz, Resende, and also the current author. In particular, we construct a series of dual adjunctions between the categories of Boolean spaces and skew Boolean algebras, unital versions of which are induced by dualizing objects ${0,1,..., n+1}$, $n\geq 0$. We describe Eilenberg-Moore categories of the monads of our adjunctions and construct easily understood non-commutative reflections of skew Boolean algebras, where the latter can be faithfully embedded (if $n\geq 1$) in a canonical way. As an application, we answer the question that arose in a recent paper by Leech and Spinks to describe the left adjoint to their `twisted product' functor $\omega$.