On set-representable orthocomplemented difference lattices
type: Item
instance of: scholarly article
title: On set-representable orthocomplemented difference lattices
zbMATH Open document ID: 1456.06009
author: Jianning Su
published in: Order
publication date: 2020-10-12T00:00:00Z
review text: An orthocomplemented difference lattice is an algebra \(\mathbf L=(L,\vee,\wedge,\Delta,{}',0,1)\) of type \((2,2,2,1,0,0)\) such that \((L,\vee,\wedge,{}',0,1)\) is an orthocomplemented lattice and the following identities are satisfied: \(\Delta\) is associative, \(x\Delta1\approx1\Delta x\approx x'\) and \(x\Delta y\le x\vee y\). \(\mathbf L\) is called set-representable if there exists a set \(X\) and a subset \(\Omega\) of \(2^X\) such that \(X\in\Omega\), \(A\Delta B\in\Omega\) for all \(A,B\in\Omega\) and \((L,\le,\Delta,0,1)\cong(\Omega,\subseteq,\Delta,\emptyset,X)\). It is proved that the class of set-representable orthocomplemented difference lattices is not locally finite. An orthomodular lattice is called concrete if it is isomorphic to a collection of subsets of a set with partial ordering given by set inclusion, orthocomplementation given by set complementation and finite orthogonal joins given by disjoint unions. An orthomodular lattice \(\mathbf M\) is said to be OML-embeddable into \(\mathbf L\) if \(\mathbf M\) can be embedded into the orthomodular lattice \((L,\vee,\wedge,{}',0,1)\). It is proved that not every concrete orthomodular lattice can be OML-embedded into an orthocomplemented difference lattice. Reviewer's remark: In the first reference ``Länger, H. M.'' should be ``Länger, H.''.
reviewed by: Helmut Länger
Mathematics Subject Classification ID: 06C15
Mathematics Subject Classification ID: 03G12
Mathematics Subject Classification ID: 81P10
zbMATH DE Number: 7259279
zbMATH Keywords: orthocomplemented difference lattice
zbMATH Keywords: orthocomplemented lattice
zbMATH Keywords: set-representable
zbMATH Keywords: locally finite
zbMATH Keywords: orthomodular lattice
zbMATH Keywords: concrete
zbMATH Keywords: OML-embeddable
MaRDI profile type: Publication
full work available at URL: s11083-020-09522-7
OpenAlex ID: W3010027235
cites work: Q4389694
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cites work: A NOTE ON SYMMETRIC DIFFERENCES OF ORTHOMODULAR LATTICES
DOI: 10.1007/S11083-020-09522-7
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description: scientific article; zbMATH DE number 7259279