Disjointness in Partially Ordered Vector Spaces

A notion of disjointness in arbitrary partially ordered vector spaces is introduced by calling two elements x and y disjoint if the set of all upper bounds of x + y and −x − y equals the set of all upper bounds of x − y and −x + y. Several elementary properties are easily observed. The question whether the disjoint complement of a subset is a linear subspace appears to be more difficult. It is shown that in directed Archimedean spaces disjoint complements are always subspaces. The proof relies on theory on order dense embedding in vector lattices. In a non-Archimedean directed space even the disjoint complement of a singleton may fail to be a subspace. According notions of disjointness preserving operator, band, and band preserving operator are defined and some of their basic properties are studied.