Pattern Recognition on Oriented Matroids

9.2Convex Sets and Tope Committees

In this section, where we follow Section 6.10 almost word for word, we continue to count tope committees for oriented matroids.

Let the antichain $\Upsilon :=\left\{{\upsilon }_1,\mathrm{...},{\upsilon }_t\right\}\subset \mathbb{B}{\left(T\right)}^{\left(\left|T\right|/2\right)}$ again represent the family of positive halfspaces of a simple oriented matroid := (Et, ), which is not acyclic, in the Boolean lattice () of all subsets of its set of topes . We have

$\begin{array}{l}|I_{\frac{1}{2},k}(\mathbb{B}(T\end{array}$