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A DUALIZING OBJECT APPROACH TO NONCOMMUTATIVE STONE DUALITY

Part of: Basic constructions Special categories

Published online by Cambridge University Press:  19 August 2013

GANNA KUDRYAVTSEVA
GANNA KUDRYAVTSEVA*
Affiliation:
Faculty of Computer and Information Science, University of Ljubljana, Tržaška cesta, 25, SI-1001, Ljubljana, Slovenia
*
ganna.kudryavtseva@fri.uni-lj.si
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Abstract

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The aim of the present paper is to extend the dualizing object approach to Stone duality to the noncommutative setting of skew Boolean algebras. This continues the study of noncommutative 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 this paper we construct a series of dual adjunctions between the categories of left-handed skew Boolean algebras and Boolean spaces, the unital versions of which are induced by dualizing objects $\{ 0, 1, \ldots , n+ 1\} $, $n\geq 0$. We describe the categories of Eilenberg-Moore algebras of the monads of the adjunctions and construct easily understood noncommutative reflections of left-handed 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 $.

Keywords

Boolean spaceBoolean algebraStone dualityskew Boolean algebraétale spacedualizing objectschizophrenic objectadjunctionmonadEilenberg–Moore algebrareflective subcategory

MSC classification

Secondary: 06E15: Stone spaces (Boolean spaces) and related structures 06E75: Generalizations of Boolean algebras 18B30: Categories of topological spaces and continuous mappings 54B40: Presheaves and sheaves
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 95 , Issue 3 , December 2013 , pp. 383 - 403
Copyright
Copyright ©2013 Australian Mathematical Publishing Association Inc. 

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