Quantales and continuity spaces
type: Item
instance of: scholarly article
title: Quantales and continuity spaces
zbMATH Open document ID: 0906.54024
author: R. G. Flagg
published in: Algebra Universalis
publication date: 1998-11-23T00:00:00Z
review text: The paper under review attempts to generalize the notion of a metric using as distance sets the so-called value quantales. Such a generalized metric space is called a continuity space. In the first part of the paper the author introduces the notion of a value distributive lattice as a completely distributive lattice such that the set of all elements well above the bottom element is a filter. In the second part the notion of a value quantale as a quantale that is a value distributive lattice is defined. He proves that any element \(p\) in a value quantale is the meet of all elements \(p+c\), \(c\) well above 0. In the next part the notion of a \(V\)-continuity space is introduced. Natural examples of \(V\)-continuity spaces are posets, (probabilistic) metric spaces and structure spaces. Since we have a generalized metric, the author defines the notion of an open ball and a closed ball. So we have an induced topology, the dual topology and the symmetric topology. The author proves that the symmetric topology is the join of the induced topology and the dual one. The last part establishes the connection between the value semigroups and the value quantales in the context of continuity spaces.
reviewed by: Jan Paseka
Mathematics Subject Classification ID: 54E99
Mathematics Subject Classification ID: 06F05
Mathematics Subject Classification ID: 06D10
zbMATH DE Number: 1226205
zbMATH Keywords: generalized metric space
zbMATH Keywords: value distributive lattice
zbMATH Keywords: continuity space
zbMATH Keywords: completely distributive lattice
zbMATH Keywords: value quantale
zbMATH Keywords: induced topology
zbMATH Keywords: value semigroups
MaRDI profile type: Publication
full work available at URL: s000120050018
OpenAlex ID: W2121615522
DOI: 10.1007/S000120050018
description: scientific article; zbMATH DE number 1226205