Epicompletion in frames with skeletal maps. IV: \(\ast\)-regular frames
type: Item
instance of: scholarly article
title: Epicompletion in frames with skeletal maps. IV: \(\ast\)-regular frames
zbMATH Open document ID: 1260.06011
published in: Applied Categorical Structures
publication date: 2012-09-27T00:00:00Z
review text: In Part I of this series of papers [Appl. Categ. Struct. 16, No. 4, 521--533 (2008; Zbl 1156.06004)], the author and \textit{E. R. Zenk} started an investigation into epicompletion in various categories of compact frames with skeletal maps. The paper under review is the fourth installment in this series of investigations. In the second installment [the author and \textit{E. R. Zenk}, ibid. 17, No. 5, 467--486 (2009; Zbl 1184.06005)], an epireflection, denoted \(\psi\), was constructed. This was an epireflection of the category of compact normal joinfit frames in the subcategory consisting of strongly projectable (also known as extremely disconnected) frames. In the third contribution in this anthology [the author, ibid. 19, No. 2, 489--504 (2011; Zbl 1223.06010)], one of the main achievements was the demonstration that the epireflection \(\psi\) is actually a monoreflection, and that the reflection map is a closed map. In this article, the author continues the investigation. He introduces \(\ast\)-regular frames -- a notion which broadens the domain of the monoreflection \(\psi\). These are frames for which the inclusion map from the regular coreflection of a frame into the frame is \(\ast\)-dense; meaning that, whenever the right adjoint of the inclusion map sends an element to the bottom, then the element is the bottom. The usual saturation nucleus is shown to be a reflection with respect to weakly closed maps, in the subcategory of subfit objects. Furthermore, the saturation nucleus commutes with \(\psi\) on compact normal joinfit frames with skeletal maps which are weakly closed. The paper closes with an illustration of the scope of the material developed in the earlier parts. This includes examples of frames that satisfy some properties, but not others. For instance, there is an example (which the author attributes to Warren McGovern) of a coherent \(\ast\)-regular frame which is not normal. Building on this example, using saturation and the monoreflection \(\p…
reviewed by: Themba Dube
Mathematics Subject Classification ID: 06D22
Mathematics Subject Classification ID: 18A20
Mathematics Subject Classification ID: 18A30
Mathematics Subject Classification ID: 18A40
Mathematics Subject Classification ID: 18B30
zbMATH DE Number: 6087663
zbMATH Keywords: epicompletion
zbMATH Keywords: skeletal map
zbMATH Keywords: \(\ast\)-regular frame
zbMATH Keywords: joinfit frame
zbMATH Keywords: saturation
zbMATH Keywords: monoreflection
MaRDI profile type: Publication
full work available at URL: s10485-009-9220-5
OpenAlex ID: W2055816306
cites work: Q3830990
cites work: Q4884914
cites work: Q4336219
cites work: Functorial maximal spectra
cites work: The essential closure of an Archimedean lattice-ordered group
cites work: Q3877831
cites work: Prime Ideal Structure in Commutative Rings
cites work: Q3965241
cites work: Archimedean lattices
cites work: Q3568664
cites work: Epicompletion in frames with skeletal maps. III: When maps are closed
cites work: When an algebraic frame is regular
cites work: Nuclear typing of frames vs spatial selectors
cites work: Epicompletion in frames with skeletal maps. I: Compact regular frames
cites work: Epicompletion in frames with skeletal maps. II: Compact normal joinfit frames
cites work: Q4459228
DOI: 10.1007/S10485-009-9220-5
Recommended article: Epicompletion in frames with skeletal maps. III: When maps are closed
Recommended article: Epicompletion in frames with skeletal maps. II: Compact normal joinfit frames
Recommended article: Epicompletion in frames with skeletal maps. I: Compact regular frames
Recommended article: Essential completeness in categories of completely regular frames.
Recommended article: Morphisms and pushouts in compact normal joinfit frames
Recommended article: \(E\)-compactness in pointfree topology
Recommended article: Pointfree functorial polar functions
Recommended article: Q4852559
Recommended article: Regular monomorphisms of Hausdorff frames
Recommended article: Compactification of \(\kappa\)-frames
author: Jorge Martínez
description: scientific article; zbMATH DE number 6087663