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

## TL;DR

This paper explores decompositions of topes in simple oriented matroids, establishing unique minimal subsets related to symmetric cycles and revealing orthogonality relations under specific conditions.

## Contribution

It introduces a novel framework for decomposing topes via symmetric cycles and uncovers orthogonality relations based on subset sizes and parity conditions.

## Key findings

- Unique minimal subsets Q(T,R) exist for topes T in simple oriented matroids.
- Orthogonality relations hold for certain decompositions with large subset sizes and opposite parity ground sets.
- The results connect to structural properties of tope graphs and symmetric cycles in 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 for decompositions Q(T',R') and Q(T",R") with respect to symmetric cycles R' and R" in the tope graphs of two simple oriented matroids, whose ground sets have the cardinalities of opposite parity, we have |Q(T',R')|>3 and |Q(T",R")|>3, then these decompositions satisfy a certain orthogonality relation.