×

Kombinatorische Realisierbarkeitskriterien für orientierte Matroide. (Criteria for the realizability of oriented matroids). (German) Zbl 0723.05032

Mitt. Math. Semin. Gießen 194, 1-112 (1989).
Summary: The realizability problem as well as the isotopy problem for oriented matroids have been studied in several papers. In the first part of this paper we compare three types of realisation algorithms: “solvability sequences”, “free solvability sequences” and “reduction sequences”. In the second part we introduce a new realisation method: the “pseudo reduction sequences”. We prove that the four classes of uniform rank-3 oriented matroids, which can be realized by these four methods, form a strictly increasing hierarchy. Furthermore we show that any uniform oriented matroid, that admits a “pseudo reduction sequence” also has the isotopy property. We use this result to show, that every realizable uniform rank-3 matroid up to nine points hat the isotopy property. This improves recently published results of S. M. Finashin [Lect. Notes Math. 1346, 501-526 (1988; Zbl 0653.14021)].

MSC:

05B35 Combinatorial aspects of matroids and geometric lattices

Citations:

Zbl 0653.14021