On the normal completion of a Boolean algebra
type: Item
instance of: scholarly article
title: On the normal completion of a Boolean algebra
zbMATH Open document ID: 1018.06008
DOI: 10.1016/S0022-4049(02)00290-6
author: Mojgan Mahmoudi
author: Bernhard Banaschewski
author: M. Mehdi Ebrahimi
published in: Journal of Pure and Applied Algebra
publication date: 2003-05-25T00:00:00Z
review text: A familiar construction for a Boolean algebra \(A\) is its normal completion \({\mathcal N}_A\) (given by its normal ideals or, equivalently, the intersections of its principal ideals, together with the embedding \(A\rightarrow {\mathcal N}_A\) taking each element of \(A\) to its principal ideal). In the classical setting of Zermelo-Fraenkel set theory with Choice, \({\mathcal N}_A\) is characterized in various ways; thus, it is the unique complete extension of \(A\) in which the image of \(A\) is join-dense, the unique essential completion of \(A\), and the injective hull of \(A\). In this paper, the authors are interested in characterizing the normal completion in the constructive context of an arbitrary topos. They show, among other things, that it is, even at this level, the unique join-dense, or alternatively, essential completion. En route, they investigate the functorial properties of \({\mathcal N}_A\) and establish that is the reflection of \(A\), in the category of Boolean homomorphisms which preserve all existing joins, to the complete Boolean algebras. In this context, they make crucial use of the notion of a skeletal frame homomorphism.
Mathematics Subject Classification ID: 06E05
Mathematics Subject Classification ID: 18B25
Mathematics Subject Classification ID: 18A40
Mathematics Subject Classification ID: 18D35
Mathematics Subject Classification ID: 08B30
Mathematics Subject Classification ID: 03F65
zbMATH DE Number: 1915687
zbMATH Keywords: Boolean algebra
zbMATH Keywords: normal completion
zbMATH Keywords: topos
zbMATH Keywords: skeletal frame homomorphism
reviewed by: Dimitru Buşneag
MaRDI profile type: Publication
cites work: Q5641296
cites work: Boolean algebras in a localic topos
cites work: Q4883943
cites work: Q4674837
cites work: Algebra in a Grothendieck topos: Injectivity in quasi-equational classes
cites work: Internal completeness and injectivity of Boolean algebras in the topos of M-sets
cites work: Q4145861
cites work: Q3965241
cites work: Q3889051
cites work: An extension of the Galois theory of Grothendieck
cites work: Sheaves in geometry and logic: a first introduction to topos theory
cites work: Internal injectivity of Boolean algebras in \({\mathcal{MS}}et\)
description: scientific article; zbMATH DE number 1915687