Compatible topologies on mixed lattice vector spaces

TL;DR

This paper develops the foundational topological theory for mixed lattice vector spaces, exploring compatible topologies, their characterizations, and connections to locally solid structures, extending classical Riesz space results.

Contribution

It introduces a characterization of compatible topologies on mixed lattice spaces, analogous to Roberts-Namioka for Riesz spaces, and explores their properties and connections.

Findings

Characterization of compatible mixed lattice topologies

Analysis of locally convex and seminormed structures

Connections between mixed lattice and locally solid topologies

Abstract

A mixed lattice vector space is a partially ordered vector space with two partial orderings, generalizing the notion of a Riesz space. Whereas the algebraic theory of mixed lattice structures dates back to the 1970s, the topological theory of mixed lattice spaces remains largely unexplored. 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 obtain a characterization of compatible mixed lattice topologies, which is similar to the well known Roberts-Namioka characterization of locally solid Riesz spaces. Moreover, we study locally convex topologies and the associated seminorms, as well as connections between mixed lattice topologies and locally solid topologies on Riesz spaces. We also briefly discuss asymmetric norms and cone norms on mixed lattice spaces.