Pattern Recognition on Oriented Matroids: Topes and Critical Committees

Abstract

Let the sign components of the maximal covectors of a simple oriented matroid M be represented by the real numbers -1 and 1. Consider the vertex set V(R) of a symmetric cycle R of adjacent topes in the tope graph of M as a subposet of the tope poset of M. If B is the bottom element of the tope poset then B is equal to the unweighted sum of the members of the set min V(R) of minimal elements of the subposet V(R); if B is the positive tope then the set min V(R) is a critical tope committee for the acyclic oriented matroid M.

7 pages, 3 figures. v2,3 - misprints corrected, Corollary 2.2 and Example 2.3 modified; v.4,5,6 - minor improvements

References in corpus (5)

• Pattern Recognition on Oriented Matroids: Layers of Tope Committees
• Pattern Recognition on Oriented Matroids: The Existence of a Tope Committee
• Pattern Recognition on Oriented Matroids: Three-Tope Committees
• Pattern Recognition on Oriented Matroids: K*-Vectors and Reorientations
• Pattern Recognition on Oriented Matroids: Halfspaces, Convex Sets and Tope Committees

Cited by in corpus (2)

• Pattern Recognition on Oriented Matroids: Symmetric Cycles in the Hypercube Graphs
• Pattern Recognition on Oriented Matroids: Critical Committees and Distance Signals