Dimension and Embedding Theorems for Geometric Lattices

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, FE G with dimE=e, dimF=f, EvF=l, and dimEhF=e+f-n-l; 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.