Compatible topologies on mixed lattice vector spaces

Abstract

A mixed lattice vector space is a partially ordered vector space with two partial orderings, generalizing the notion of a Riesz space. The purpose of this paper is to develop the basic topological theory of mixed lattice spaces. A vector topology is said to be compatible with the mixed lattice structure if the mixed lattice operations are continuous. We give a characterization of compatible mixed lattice topologies, similar to the well known Roberts-Namioka characterization of locally solid Riesz spaces. We then study locally convex topologies and the associated seminorms, as well as connections between mixed lattice topologies and locally solid topologies on Riesz spaces. In the locally convex case, we obtain a more complete characterization of compatible mixed lattice topologies. We also briefly discuss asymmetric norms and cone norms on mixed lattice spaces with a particular application to finite dimensional spaces.