Designing software based on oriented matroid Theory

Abstract

This work presents the design of C++ software aimed at the analysis of oriented matroids, a fundamental structure in combinatorial geometry that extends the concept of linear independence. Key notions of the theory are addressed, such as bases, circuits, uniform matroids, and the chirotope (χ), which describes the orientation of the matroid. The development is based on a formula proposed by Dr. J. A. Nieto to construct a matrix representing an oriented matroid from n elements and its corresponding chirotope. This computational implementation generates square matrices of size n x n, but analysis shows that, in general, these do not satisfy the necessary conditions of linear dependence expected in matroids of rank, as they yield nonzero determinants, contradicting their validity as representations over. As an emergent result, it is proposed that the submatrix formed by the first rows of the generated matrix may constitute a valid representation of the matroid, a hypothesis supported by computational examples. It is concluded that the software represents a functional advance in the computational study of oriented matroids, and future lines of research are identified to formally validate the base formula, optimize its algorithmic complexity, and characterize the computational representability of such structures.