Ortho and causal closure operations in ordered vector spaces
type: Item
instance of: scholarly article
title: Ortho and causal closure operations in ordered vector spaces
zbMATH Open document ID: 1161.06003
published in: Algebra Universalis
publication date: 2009-01-29T00:00:00Z
review text: Let \(V\) be a partially ordered real vector space. The orthogonality relation \(\perp\) is defined by incomparability. Two closure operators on the subsets of~\(V\) are defined and studied: orthoclosure \(A\mapsto A^{\perp\perp}\) and causal closure \(A\mapsto D(A)= \{a\in V \mid \forall b\in V\), \(b>0 \Rightarrow \exists \omega\in\mathbb{R}\), \(a+\omega b\in A\}\). It is known that the double orthoclosed sets form a complete orthocomplemented poset. This poset is orthomodular if \(V\) is integrally open, i.e., the conjunction of \(b > 0\) and \(a < b\) implies \(\exists n \in N\), \(a + nb > 0\). The inclusion \(D(A)\subseteq A^{\perp\perp}\) always holds. The main result is that \(V\) is integrally open iff the two closure operators coincide on orthogonal sets.
reviewed by: Mirko Navara
Mathematics Subject Classification ID: 06C15
Mathematics Subject Classification ID: 06F20
Mathematics Subject Classification ID: 06A15
zbMATH DE Number: 5500444
zbMATH Keywords: ordered vector space
zbMATH Keywords: orthogonality space
zbMATH Keywords: orthomodular lattice
author: Jan Florek
MaRDI profile type: Publication
full work available at URL: s00012-008-2088-7
OpenAlex ID: W2015184608
DOI: 10.1007/S00012-008-2088-7
description: scientific article; zbMATH DE number 5500444