A finite crisscross method for oriented matroids

Fachbereich 3 Mathematik Oriented Matroids Oriented Matroids

Oriented Matroids

The theory of oriented matroids provides a broad setting in which to model, describe, and analyze combinatorial properties of geometric configurations. Apparently totally different mathematical objects such as point and vector configurations, arrangements of hyperplanes, convex polytopes, directed graphs, and linear programs find a common generalization in the language of oriented matroids. The...

On a Mutation Problem for Oriented Matroids

For uniform oriented matroids M with n elements, there is in the realizable case a sharp lower bound Lr (n) for the number mut(M) of mutations ofM : Lr (n) = n ≤ mut(M), see Shannon [17]. Finding a sharp lower bound L(n) ≤ mut(M) in the non-realizable case is an open problem for rank d ≥ 4. Las Vergnas [9] conjectured that 1 ≤ L(n). We study in this article the rank 4 case. RichterGebert [11] s...

On exchange properties for Coxeter matroids and oriented matroids

We introduce new basis exchange axioms for matroids and oriented matroids. These new axioms are special cases of exchange properties for a more general class of combinatorial structures, Coxeter matroids. We refer to them as “properties” in the more general setting because they are not all equivalent, as they are for ordinary matroids, since the Symmetric Exchange Property is strictly stronger ...

Shellability of Oriented Matroids

In Man82] A. Mandel proved that the maximal cells of an Oriented Matroid poset are B-shellable. Our result shows that the whole Oriented Matroid is shellable, too.