Séminaire Lotharingien de Combinatoire, B18b (1987).
[Formerly: Publ. I.R.M.A. Strasbourg, 1988, 358/S-18, p. 137-138.]

Andreas Dress and Walter Wenzel

Matroids with Coefficients

Abstract. One of the most important branches in matroid theory is to represent matroids over fields -- this means essentially to embed combinatorial geometries into vector spaces. A further important concept in this context was introduced by R.G. Bland and M. Las Vergnas, namely the concept of oriented matroids. These may be viewed as an abstraction of matroids representable over the real field, though allowing for examples which are not representable over any field (or ring). It turned out that the study of representability and orientability of matroids showed very similar aspects. In view of this phenomenon, A. Dress and the author were motivated to establish a theory of matroids with coefficients in a more general coefficient domain, called a fuzzy ring, by weakening the axioms of a ring, thereby unifying the theories of representable and orientable matroids. This has been possible in view of the well-known Grassmann Plücker identity for determinants. In the case of oriented matroids one gets an appropriate coefficient domain by "identifying" in the real field all numbers of the same sign. In this talk, an introduction into the theory of matroids with coefficients in a fuzzy ring is given.

This paper is a summary of:

A.W.M. Dress: Duality Theory for Finite and Infinite Matroids with Coefficients, Advances in Mathematics 59 (1986), 97-123.

A.W.M. Dress, W. Wenzel: Grassmann-Plücker Relations and Matroids with Coefficients, Advances in Mathematics 86 (1991), 68-110.