A UNIFYING INTERPRETATION OF SEVERAL COMBINATORIAL DUALITIES (DUALITY, GREEDOIDS, MATROIDS)

摘要: Several combinatorial structures have an interesting duality relation. Examples include matroid duality, the blocking duality of clutters, oriented matroid duality, and weakly oriented matroid duality. We show that, in a sense, these duality relations are the same; only the setting in which they act changes. We do this by giving a short list of properties that, when applied to each structure, characterizes its duality relation.Vector space orthogonality can be regarded to be a duality relation for subspaces of finite dimensional vector spaces F (''E) over a field F. This relation satisfies the fundamental properties used to characterize the combinatorial dualities. In fact, for the field of real numbers, the field of rational numbers, and the fields GF (p (''n)) where p is a prime and n is odd, these properties characterize vector space orthogonality.