On Circuit Valuation of Matroids

The concept of valuated matroids was introduced by Dress and Wenzel as a quantitative extension of the base exchange axiom for matroids. This paper gives several sets of cryptomorphically equivalent axioms of valuated matroids in terms of (R∪{-∞})-valued vectors defined on the circuits of the underlying matroid, where R is a totally ordered additive group. The dual of a valuated matroid is characterized by an orthogonality of (R∪{-∞})-valued vectors on circuits. Minty's characterization for matroids by the painting property is generalized for valuated matroids.