The dimension theorem in axiomatic geometry

Abstract

Given a matroid and an integer n ≥ 0, eleven conditions are shown to be equivalent to the validity of the rank formula r(E ∧ F) + r(E ∨ F = r(E) + r(F) for subspaces satisfying r(E∧F)≥n. For n=0 one finds the projective geometries. The case n=1 also includes the affine and the hyperbolic geometries, the case n=2 the Mobius geometries. The general case covers the incidence geometries of grade n of Wille.