Some ring-theoretic properties of almost \(P\)-frames
type: Item
instance of: scholarly article
title: Some ring-theoretic properties of almost \(P\)-frames
zbMATH Open document ID: 1186.06006
author: Themba Dube
published in: Algebra Universalis
publication date: 2010-01-04T00:00:00Z
review text: An almost \(P\)-frame is a frame \(L\) in which every cozero element is regular (that is, for any cozero element \(a\) of \(L\), if the pseudocomplement of \(a\) is the bottom element of \(L\), then \(a\) is necessarily the top element of \(L\)) [\textit{R. N. Ball} and \textit{J. Walters-Wayland}, ``\(C\)- and \(C^*\)-quotients in pointfree topology'', Diss. Math. 412, 62 p. (2002; Zbl 1012.54025)]. Almost \(P\)-frames generalize almost \(P\)-spaces in the sense that for any topological space \((X,{\mathfrak O}X)\), \((X,{\mathfrak O}X)\) is an almost \(P\)-space if and only if the frame \({\mathfrak O}X\) of its open sets is an almost \(P\)-frame. This paper provides several characterizations of almost \(P\)-frames (and \(P\)-frames). They enable the author to give, in the appendix, frame versions of the theorem of Gelfand and Kolmogorov concerning maximal ideals of the ring \(C(X)\), and of a lemma due to J. D. McKight jun., which tells us where all the ideals of \(C(X)\) lie.
reviewed by: Jorge Picado
Mathematics Subject Classification ID: 06D22
Mathematics Subject Classification ID: 13A15
Mathematics Subject Classification ID: 54C40
Mathematics Subject Classification ID: 54G05
zbMATH DE Number: 5652274
zbMATH Keywords: frame of reals
zbMATH Keywords: ring of continuous real functions
zbMATH Keywords: \(P\)-frame
zbMATH Keywords: almost \(P\)-frame
zbMATH Keywords: ideal
MaRDI profile type: Publication
full work available at URL: s00012-009-2093-5
OpenAlex ID: W2106682579
DOI: 10.1007/S00012-009-2093-5
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description: scientific article; zbMATH DE number 5652274