Abstract :

We study pre-Riesz spaces, i.e. partially ordered vector spaces that embed order densely into vector lattices. Concepts from vector lattice theory such as disjointness, ideals, and bands are extended to pre-Riesz spaces, as well as disjointness preserving operators and related notions. The analysis of these concepts revolves around embedding techniques into vector lattice covers, in particular into the functional representation. Since every directed Archimedean partially ordered vector space is a pre-Riesz space, every ordered Banach space with generating closed cone is as well. We give examples of results from the theory of Banach lattices, for instance concerning disjointness preserving operators, that can smoothly be generalized to ordered Banach spaces. On the other hand, we give instances where the vector lattice theory and the theory of pre-Riesz spaces differ essentially.