Disjointness in partially ordered vector spaces
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.
scientific article; zbMATH DE number 5070787
scientific article