Boolean quotients of orthomodular lattices
type: Item
instance of: scholarly article
title: Boolean quotients of orthomodular lattices
zbMATH Open document ID: 0840.03050
DOI: 10.1007/BF01181873
author: Sylvia Pulmannová
author: Anna Bruna d'Andrea
published in: Algebra Universalis
publication date: 1996-01-23T00:00:00Z
review text: Let \(L\) be an orthomodular lattice and \(J\) a proper \(p\)-ideal of \(L\) (i.e. a lattice ideal such that \(a \in L\), \(b \in L \Rightarrow (a \vee b') \wedge b \in L)\). The present paper investigates the properties of those orthomodular lattices for which there exist nontrivial Boolean quotients \(L/J\). In particular, the following result is shown: If \(G\) is any commutative group not of order 2, then \(L\) has a nontrivial Boolean quotient if and only if there exists a function \(m : L \to \{0, \lambda\}\), where \(\lambda \in G\) is not of order 2, such that \(m (a \vee b) + m(a \wedge b) = m(a) + m(b)\) for any \(a\), \(b \in L\). This implies a generalization to commutative groups of a well-known no-go theorem concerning the nonexistence of dispersion-free states in Quantum Mechanics. The authors also construct an example of a non-Boolean orthomodular lattice with all nontrivial quotients Boolean.
Mathematics Subject Classification ID: 03G12
Mathematics Subject Classification ID: 06C15
Mathematics Subject Classification ID: 81P10
zbMATH DE Number: 836829
zbMATH Keywords: orthomodular lattice
zbMATH Keywords: Boolean quotients
zbMATH Keywords: dispersion-free states
reviewed by: Nicolas Hadjisavvas
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01181873
OpenAlex ID: W2077329962
description: scientific article; zbMATH DE number 836829