Mutations and (Non-)Euclideaness in oriented matroids

TL;DR

This paper explores the properties of oriented matroids related to Euclidean extensions, mutations, and non-Euclidean examples, providing new classifications, bounds, and examples for these mathematical structures.

Contribution

It introduces the classes Mandel and Las Vergnas of oriented matroids, establishes proper inclusions among various classes, and provides explicit examples and bounds for mutations and Euclidean properties.

Findings

All class inclusions are proper, with explicit proofs and examples.

For realizable hyperplane arrangements, L equals the rank, confirming Shannon's result.

Euclidean oriented matroids require at least three mutations to reach non-Euclidean ones.

Abstract

We call an oriented matroid Mandel if it has an extension in general position which makes all programs with that extension Euclidean. If $L$ is the minimum number of mutations adjacent to an element of the groundset, we call an oriented matroid Las Vergnas if $L > 0$. If $\frak{O}_{\mathcal{property}}$ is the class of oriented matroids having a certain property, it holds $\frak{O} \supset \frak{O}_{\mathcal{Las Vergnas}} \supset \frak{O}_{\mathcal{Mandel}} \supset \frak{O}_{\mathcal{Euclidean}} \supset \frak{O}_{\mathcal{realizable}}.$ All these inclusions are proper, we give explicit proofs/examples for the parts of this chain that were not known. For realizable hyperplane arrangements of rank $r$ we have $L = r$ which was proved by Shannon. Under the assumption that a (modified) intersection property holds we give an analogon to Shannons proof and show that uniform rank $4$ Euclidean oriented matroids with that property have $L = 4$. Using the fact that the lexicographic extension creates and destroys certain mutations, we show that for Euclidean oriented matroids holds $L \ge 3$. We give a survey of preservation of Euclideaness and prove that Euclideaness remains after a certain type of mutation-flips. This yields that a path in the mutation graph from a Euclidean oriented matroid to a totally non-Euclidean oriented matroid (which has no Euclidean oriented matroid programs) must have at least three mutation-flips. Finally, a minimal non-Euclidean or rank $4$ uniform oriented matroid is Mandel if it is connected to a Euclidean oriented matroid via one mutation-flip, hence we get many examples for Non-Euclidean but Mandel oriented matroids and have $L \le 3$ for those of rank $4$.