Matroids with coefficients over projective planes
Coordination of a projective plane \(\Pi\) by a given frame leads to a ternary field \(T\). The author considers the ``extended radical'' \(R_ a\) of \(T\) and shows that the quotient structure \(K = T/R_ a\) is a ``fuzzy integral domain'' (in the sense of A. Dress), which is independent of the chosen frame. Thus \(\Pi\) can be considered as a matroid with coefficients in \(K\), and all chirotope structures and valuations of \(\Pi\) can be recognized via \(K\). In particular, a complete survey over the valuations of \(\Pi\) is given, based on the notion of ``uniform valuation''. The multiplicative group of units of \(K\) equals the inner Tutte group of \(\Pi\), which is closely related to Junker's theory of order functions in projective planes. This is used in a more computational approach to these groups.
scientific article; zbMATH DE number 665826