Pattern Recognition on Oriented Matroids: Topes and Critical Committees

TL;DR

This paper explores the structure of oriented matroids by analyzing topes and critical committees, revealing how minimal elements in certain subposets relate to fundamental components of the matroid's tope graph.

Contribution

It introduces a novel connection between minimal elements of subposets in the tope graph and critical tope committees in simple oriented matroids.

Findings

Minimal elements of subposets sum to the bottom element of the tope poset.

Critical tope committees correspond to minimal elements in specific subposets.

Structural insights into the tope graph of oriented matroids.

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.