Singly generated frame extensions
type: Item
instance of: scholarly article
title: Singly generated frame extensions
zbMATH Open document ID: 0774.18004
DOI: 10.1016/0022-4049(92)90101-K
published in: Journal of Pure and Applied Algebra
publication date: 1993-05-16T00:00:00Z
review text: As the title suggests, this paper investigates frames which arise from a given frame by adjoining a single element. More precisely, a frame \(M\) is a singly generated frame extension of a frame \(L\) if and only if \(L\) is a subframe of \(M\) and \(M\) is generated by \(L\) together with \(c\in M\). Such extensions are analyzed using congruences (equivalently nuclei) on the subframe \(P=\{(x,y)\mid x\leq y\}\) of \(L\times L\). This analysis includes a characterization of epi-extensions, that is, when the identity embedding \(L\to M\) is an epimorhism. The rest of the article considers frame theoretic separation properties, in particular the axiom of strong Hausdorffness (called ``separated'' here) due to \textit{J. R. Isbell} [Math. Scand. 31, 5-32 (1972; Zbl 0246.54028)]. It is first considered in the context of singly generated extensions, and then an appendix looks at it relative to more general extensions.
reviewed by: K. I. Rosenthal
Mathematics Subject Classification ID: 18B30
Mathematics Subject Classification ID: 54B30
zbMATH DE Number: 166149
zbMATH Keywords: frame extension
zbMATH Keywords: congruences
zbMATH Keywords: nuclei
zbMATH Keywords: epi-extensions
zbMATH Keywords: separation
zbMATH Keywords: axiom of strong Hausdorffness
author: Bernhard Banaschewski
MaRDI profile type: Publication
cites work: Extensions of Topological Spaces
cites work: Q5522002
cites work: Q4888143
cites work: An asymmetric characterization of the congruence frame
cites work: Atomless Parts of Spaces.
cites work: Q3711309
cites work: Epimorphisms of frames
cites work: On $H$-closed extensions of topological spaces
cites work: The lattice theoretic part of topological separation properties
description: scientific article; zbMATH DE number 166149