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

## TL;DR

This paper explores the decomposition of topes in oriented matroids using symmetric cycles, revealing unique minimal subsets and establishing Dehn-Sommerville type relations for complex decompositions.

## Contribution

It introduces a novel decomposition method for topes in oriented matroids and connects these decompositions to Dehn-Sommerville type relations, advancing theoretical understanding.

## Key findings

- Unique minimal subsets Q(T;R) exist for topes T
- Decompositions with more than three elements satisfy specific relations
- Provides new insights into the structure of oriented matroids

## 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)|>3, then the decomposition Q(T;R) of the tope T with respect to the symmetric cycle R satisfies certain Dehn-Sommerville type relations.