Bands in partially ordered vector spaces with order unit

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 $n\geq 4$.