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 $$n\ge 4$$ .