Observables on lexicographic effect algebras
type: Item
instance of: scholarly article
title: Observables on lexicographic effect algebras
zbMATH Open document ID: 1445.03074
author: Anatolij Dvurečenskij
author: Dominik Lachman
published in: Algebra Universalis
publication date: 2019-11-27T00:00:00Z
review text: For the notion of lexicographic effect algebras, see the paper [Algebra Univers. 75, No. 4, 451--480 (2016; Zbl 1350.03046)] by the first author. In the paper under review, lexicographic effect algebras are considered that are intervals in lexicographic products \(H \overrightarrow\times G\), where \(\langle H, u \rangle\) is a unital po-group and \(G\) is a monotone \(\sigma\)-complete po-group with interpolation. An observable on such an effect algebra \(E\) is a kind of \(\sigma\)-homomorphism from the Borel algebra \(\mathcal{B}(\mathbb R)\) into \(E\). A spectral resolution on \(E\) is a certain system of elements \(x_t \in E\) with \(t \in \mathbb R\), which is, in particular, monotonic, left continuous in a sense and has \(0\) as the infimum and \(1\) as the supremum. Every observable \(x\) provides a spectral resolution where \(x_t :=x ((-\infty,t))\); this resolution posseses the so called finiteness property. The main result states that, conversely, every spectral resolution on \(E\) having this property is induced in this way by a unique observable. Several other properties of spectral resolutions are also discussed; there is also a number of illustrating examples.
reviewed by: Jānis Cīrulis
Mathematics Subject Classification ID: 03G12
Mathematics Subject Classification ID: 06C15
Mathematics Subject Classification ID: 06F15
Mathematics Subject Classification ID: 81P10
zbMATH DE Number: 7137282
zbMATH Keywords: effect algebra
zbMATH Keywords: finiteness property
zbMATH Keywords: lexicographic effect algebra
zbMATH Keywords: monotone σ-complete po-group
zbMATH Keywords: observable, spectral resolution
MaRDI profile type: Publication
full work available at URL: s00012-019-0628-y
OpenAlex ID: W2986725213
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Wikidata QID: Q126800152
DOI: 10.1007/S00012-019-0628-Y
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description: scientific article; zbMATH DE number 7137282