Dimension and embedding theorems for geometric lattices

Abstract

Let G be an n-dimensional geometric lattice. Suppose that 1 ⩽ e, f ⩽ n − 1, e + f ⩾ n, but e and f are not both n − 1. Then, in general, there are E, F ϵ G with dim E = e, dim F = f, E ∇ F = 1, and dim E ∧ F = e + f − n − 1; any exception can be embedded in an n-dimensional modular geometric lattice M in such a way that joins and dimensions agree in G and M, as do intersections of modular pairs, while each point and line of M is the intersection (in M) of the elements of G containing it.