Concerning \(P\)-frames, essential \(P\)-frames, and strongly zero-dimensional frames
type: Item
instance of: scholarly article
title: Concerning \(P\)-frames, essential \(P\)-frames, and strongly zero-dimensional frames
zbMATH Open document ID: 1190.06007
author: Themba Dube
published in: Algebra Universalis
publication date: 2009-11-25T00:00:00Z
review text: A topological space is said to be a \(P\)-space if every zero-set is open. This notion extends to the pointfree setting of frames in the following way: a frame \(L\) is a \(P\)-frame in case \(a\vee a^*=1\) for every cozero element \(a\) of \(L\). Indeed, a topological space \(X\) is a \(P\)-space if and only if the frame of open sets of \(X\) is a \(P\)-frame. In the present paper, the author introduces a generalization of \(P\)-frames, called essential \(P\)-frames, proves that they are always strongly zero-dimensional and presents several nice and useful characterizations of \(P\)-frames, essential \(P\)-frames and strongly zero-dimensional frames in terms of algebraic properties of the ring of continuous functions on the frame.
reviewed by: Jorge Picado
Mathematics Subject Classification ID: 06D22
Mathematics Subject Classification ID: 54C30
Mathematics Subject Classification ID: 54G10
zbMATH DE Number: 5638424
zbMATH Keywords: essential \(P\)-frame
zbMATH Keywords: strongly zero-dimensional frame
zbMATH Keywords: ring of continuous functions on a frame
zbMATH Keywords: ring ideal
zbMATH Keywords: \(m\)-topology
MaRDI profile type: Publication
full work available at URL: s00012-009-0006-2
OpenAlex ID: W2026314438
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DOI: 10.1007/S00012-009-0006-2
description: scientific article; zbMATH DE number 5638424