Mutations and (Non-)Euclideaness in oriented matroids

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 $4$.