Pattern Recognition on Oriented Matroids: Decompositions of Topes, and Dehn-Sommerville Type Relations

Abstract

If V(R) is the vertex set of a symmetric cycle R in the tope graph of a simple oriented matroid M, then for any tope T of M there exists a unique inclusion-minimal subset Q(T, R) of V(R) such that T is the sum of the topes of Q(T, R).If |Q(T, R)| ≥ 5, then the decomposition Q(T, R) of the tope T with respect to the symmetric cycle R satisfies certain Dehn-Sommerville type relations.

Pattern Recognition on Oriented Matroids: Decompositions of Topes, and Orthogonality Relations

“…where S(T ′′ , Q ′′ ) is the separation set of the topes T ′′ and Q ′′ ; see e.g. [5] on such complexes. Associate with the complex Λ ′′ its "long" f -vector…”