Bands in partially ordered vector spaces with order unit

TL;DR

This paper characterizes bands in Archimedean directed partially ordered vector spaces with an order unit, relates them to subsets of a compact space, and explores their extensions and bounds in finite dimensions.

Contribution

It provides a characterization of bands via subsets of the representing space and establishes bounds on the number of bands in finite-dimensional spaces.

Findings

Number of bands in n-dimensional spaces is bounded by  2^{2^n}/4 for n  2

Constructs examples with more bands than vector lattices in certain dimensions

Shows relationships between bands, their extensions, and carriers in the space

Abstract

In an Archimedean directed partially ordered vector space $X$ one can define the concept of a band in terms of disjointness. Bands can be studied by using a vector lattice cover $Y$ of $X$. If $X$ has an order unit, $Y$ can be represented as $C(\Omega)$, where $\Omega$ is a compact Hausdorff space. We characterize bands in $X$, and their disjoint complements, in terms of subsets of $\Omega$. We also analyze two methods to extend bands in $X$ to $C(\Omega)$ and show how the carriers of a band and its extensions are related. We use the results to show that in each $n$-dimensional partially ordered vector space with a closed generating cone, the number of bands is bounded by $\frac{1}{4}2^{2^n}$ for $n\geq 2$. We also construct examples of $(n+1)$-dimensional partially ordered vector spaces with ${2n\choose n}+2$ bands. This shows that there are $n$-dimensional partially ordered vector spaces that have more bands than an $n$-dimensional Archimedean vector lattice when $n\geq 4$.