Matroid Signatures Coordinatizable over a Semiring

Matroids with coefficients, recently introduced by Dress [4], generalize ordinary matroids, Tutte representations of matroids coordinatizable over a given field and oriented matroids. Oriented matroids can be defined in terms of circuit signatures [2]. In this paper we show that there are exactly four classes of circuit signatures coordinatizable, in the sense of Dress, over a semiring R the unit group of which is {+, −}, such that R is generated by {+, −} . In particular, we show that weakly oriented matroids [1, 7] are matroids with coefficients in a suitable semiring