Boolean representations of fuzzy quantum spaces
type: Item
instance of: scholarly article
title: Boolean representations of fuzzy quantum spaces
zbMATH Open document ID: 0915.03049
DOI: 10.1016/S0165-0114(96)00016-4
author: Mirko Navara
published in: Fuzzy Sets and Systems
publication date: 1999-06-27T00:00:00Z
review text: A fuzzy quantum space on a set \(X\) is a set \(F \subset [0,1]^X\) satisfying (i) the constant zero function belongs to \(F\), (ii) if \(a\in F\), then \(1-a \in F\), (iii) if \(\{a_i\}\) is a sequence of elements of \(F\), then \(\sup_i a_i \in F\), and (iv) the constant function 1/2 is not an element of \(F\). A Boolean representation of \(F\) is a Boolean \(\sigma\)-algebra \(\overline F\) with a surjective mapping \(h: \overline F \to F\) with some special conditions on the observables and states on \(F\) and on \(\overline F\). In the paper under review it is shown that (1) such a representation exists, (2) there exists a minimal and maximal representation, and (3) a minimal representation provides the uniqueness property. These representations make it possible to transfer many known result from Boolean \(\sigma\)-algebras to fuzzy quantum spaces.
reviewed by: Anatolij Dvurečenskij
Mathematics Subject Classification ID: 03G12
Mathematics Subject Classification ID: 81P10
Mathematics Subject Classification ID: 28E10
zbMATH DE Number: 1255748
zbMATH Keywords: fuzzy quantum space
zbMATH Keywords: state
zbMATH Keywords: observable
zbMATH Keywords: \(\sigma\)-observable
zbMATH Keywords: Boolean representation
zbMATH Keywords: Boolean \(\sigma\)-algebras
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1255748