Pattern Recognition on Oriented Matroids: Symmetric Cycles in the Hypercube Graphs. V

TL;DR

This paper investigates the structure of topes in oriented matroids related to hypercube graphs, focusing on symmetric cycles and their interpretations in subset families, blocking sets, and clutters.

Contribution

It introduces a novel interpretation of decompositions of topes in oriented matroids using symmetric cycles in hypercube graphs, connecting combinatorial structures.

Findings

Decomposition of topes linked to symmetric cycles in hypercube graphs.

Connections established between topes, subset families, and blocking sets.

New insights into the combinatorial structure of oriented matroids.

Abstract

We consider decompositions of topes of the oriented matroid realizable as the arrangement of coordinate hyperplanes in $\mathbb{R}^{2^t}$, with respect to a distinguished symmetric $2\cdot 2^t$-cycle in its hypercube graph of topes $\boldsymbol{H}(2^t,2)$. We seek interpretations of such decompositions in the context of subset families on the ground set $E_t:=\{1,\ldots,t\}$ and of the families of their blocking sets, in the context of clutters on $E_t$ and of their blockers.