Ideals, bands and direct sum decompositions in mixed lattice vector spaces

TL;DR

This paper explores the structure of mixed lattice vector spaces, focusing on ideals, bands, and their decompositions, extending the theory of Riesz spaces with new fundamental results and representation methods.

Contribution

It introduces new structural results for mixed lattice vector spaces, including ideal and band properties, and provides conditions for their direct sum decompositions.

Findings

Fundamental properties of ideals and bands in mixed lattice spaces

Representation of spaces as direct sums of disjoint bands

Conditions for decompositions via order projections

Abstract

A mixed lattice vector space is a partially ordered vector space with two partial orderings and certain lattice-type properties. In this paper we first give some fundamental results in mixed lattice groups, and then we investigate the structure theory of mixed lattice vector spaces, which can be viewed as a generalization of the theory of Riesz spaces. More specifically, we study the properties of ideals and bands in mixed lattice spaces, and the related idea of representing a mixed lattice space as a direct sum of disjoint bands. Under certain conditions, these decompositions can also be given in terms of order projections.