Séminaire Lotharingien de Combinatoire, B18b (1987).
[Formerly: Publ. I.R.M.A. Strasbourg, 1988, 358/S-18, p.
Andreas Dress and Walter Wenzel
Matroids with Coefficients
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.