Euclidean intersection properties (Q1089346)

There are matroids which have Euclidean and non-Euclidean orientations and there are also matroids whose inherent structure does not allow any Euclidean orientation. In this paper we discuss some lattice theoretic properties of matroids which when used in an oriented version guarantee Euclideanness. These properties depend all on the existence of intersections of certain flats (which is equivalent to Euclideanness interpreted in the Las Vergnas notation of oriented matroids). We introduce three classes of matroids having various intersection properties and show that two of them cannot be characterized by excluding finitely many minors.