Disjointness in partially ordered vector spaces
type: Item
instance of: scholarly article
title: Disjointness in partially ordered vector spaces
zbMATH Open document ID: 1109.46006
author: O. W. van Gaans
author: Anke Kalauch
published in: Positivity
publication date: 2006-11-03T00:00:00Z
review text: If \(X\) is a vector lattice, \(x,y\in X\) and \(| x| \wedge| y| =0\), then \(x\) and \(y\) are disjoint. This instance of disjointness amounts to the equality \(| x+y| =| x-y| \). In other words, any upper bound of \(x+y\) and \(-x-y\) is an upper bound of \(x-y\) and \(y-x\). The latter property does not involve the lattice structure of \(X\). The authors take it as the definition of disjointness in an arbitrary ordered vector space. The notion of disjointness leads to the natural concept of the disjoint complement of a subset \(M\) of~\(X\). The authors study the relevant analogues of the objects and properties that are tied with disjointness, among them order dense subspaces of vector lattices, directed ordered vector spaces, bands in these spaces, and band preserving operators. The article is full of elementary but relevant examples.
Mathematics Subject Classification ID: 46A40
zbMATH DE Number: 5070787
zbMATH Keywords: band preserving operator
zbMATH Keywords: disjoint complement
zbMATH Keywords: disjointness
zbMATH Keywords: disjointness preserving operator
zbMATH Keywords: order dense subspace
zbMATH Keywords: partially ordered vector space
zbMATH Keywords: pre-Riesz space
zbMATH Keywords: Riesz completion
reviewed by: S. S. Kutateladze
MaRDI profile type: Publication
full work available at URL: s11117-005-0015-0
OpenAlex ID: W2025394619
cites work: Inverses of disjointness preserving operators
cites work: Q4186829
cites work: Q3748810
cites work: Q3139959
cites work: Seminorms on ordered vector spaces that extend to Riesz seminorms on larger Riesz spaces
cites work: Q5638775
cites work: Banach lattices
DOI: 10.1007/S11117-005-0015-0
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description: scientific article; zbMATH DE number 5070787