Pattern Recognition on Oriented Matroids: Subtopes and Decompositions of (Sub)topes

TL;DR

This paper investigates the structure of topes and subtopes in simple oriented matroids with symmetric cycles, providing decompositions related to the edges of these cycles to enhance understanding of their combinatorial properties.

Contribution

It introduces novel decomposition methods for topes and subtopes based on symmetric cycles in the tope graph of oriented matroids, advancing combinatorial analysis techniques.

Findings

Decomposition formulas for topes and subtopes derived from symmetric cycles

Characterization of subtopes associated with edges of symmetric cycles

Enhanced understanding of the combinatorial structure of oriented matroids

Abstract

For a symmetric 2t-cycle in the tope graph of a simple oriented matroid M on the ground set {1,...,t}, where t is even, we describe decompositions of topes and subtopes of M with respect to the subtopes corresponding to the edges of the symmetric cycle.