A note on disjointness and discrete elements in partially ordered vector spaces
Jani Jokela
TL;DR
The paper develops $D$-disjointness and $D$-discrete elements as generalizations of classical disjointness and discreteness to partially ordered vector spaces, focusing on Archimedean and pre-Riesz contexts. It relates these notions to standard disjointness, proving $D$-disjointness implies disjointness and identifying conditions under which they coincide, particularly via the Riesz decomposition property. It shows that atoms are always $D$-discrete, and, in Archimedean spaces with an $M$-cone, atoms coincide with $D$-discrete elements; it also provides finite-dimensional existence results for $D$-discrete elements in faces and offers counterexamples to delineate the boundaries of the theory. The work closes with open questions about when the positive cone forms an $M$-cone and the broader implications for weakly pervasive pre-Riesz spaces, highlighting potential cone-based characterizations of Archimedean pre-Riesz spaces.
Abstract
The notions of disjointness and discrete elements play a prominent role in the classical theory of vector lattices. There are at least three different generalizations of the notion of disjointness to a larger class of partially ordered vector spaces. In recent years, one of these generalizations has been widely studied in the context of pre-Riesz spaces. The notion of $D$-disjointness is the most general of the three disjointness concepts. In this paper we study $D$-disjointness and the related concept of a $D$-discrete element. We establish some basic properties of $D$-discrete elements in Archimedean partially ordered spaces, and we investigate their relationship to discrete elements in the theory of pre-Riesz spaces.