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 a subspace of $$C(\Omega )$$ , 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 $$n\ge 2$$ . We also construct examples of $$(n+1)$$ -dimensional partially ordered vector spaces with $$\left( {\begin{array}{c}2n n\end{array}}\right) +2$$ bands. This shows that there are n-dimensional partially ordered vector spaces that have more bands than an n-dimensional Archimedean vector lattice when $$n\ge 4$$ .
Highlights
Bands play an important role in the theory of vector lattices
Together with concepts such as ideals and disjointness they are widely used to study the structure of vector lattices
Bands and disjointness are closely related, as in Archimedean vector lattices bands are exactly those subspaces that coincide with their double disjoint complements
Summary
Bands play an important role in the theory of vector lattices. Together with concepts such as ideals and disjointness they are widely used to study the structure of vector lattices. In [9] a functional representation method provides a vector lattice cover of an Archimedean partially ordered vector space X with unit by embedding X into a space of continuous functions C( ), where is a compact Hausdorff space This raises the question whether the bands in X can be identified in terms of subsets of. In a directed Archimedean partially ordered vector space X every band is an ideal [6, Theorem 5.14], and for a subset B ⊆ X the set Bd is a band [5, Proposition 5.5 (ii)]. Recall that a linear map : U → V , where U and V are partially ordered vector spaces, is called bipositive if for every x ∈ U one has x ≥ 0 if and only if (x) ≥ 0.
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