Envelopes of Geometric Lattices

A categorical embedding theorem is proved for geometric lattices. This states roughly that, if one wants to consider only those embeddings into pro-jective spaces having a suitable universal property, then the existence of such an embedding can be checked by seeing whether corresponding properties hold for many small intervals. Tutte's embedding theorem for binary geometric lattices is a consequence of this result. The purpose of this paper is to prove a somewhat technical theorem which provides a sufficient condition in order that a geometric lattice G be embeddable in a finite-dimensional projective space over a given field K. The condition involves all the small dimensional intervals of G containing 1. Instead of considering arbitrary embeddings, we restrict our attention to those having a universal property which implies that all embeddings of G in a finite-dimensional projective K-space are essentially the same. In order to state the theorem, we require some terminology. DEFINITIONS. 1. The dimension dim X of an element X of a geometric lattice is one less than the size of a basis of X. (This is the concept of dimension used in projective geometry: planes are 2-dimensional.) 2. An isometry from a geometric lattice G to a geometric lattice H is an order-monomorphism i: G-+ H, mapping 1 to 1, preserving the dimension of each element of dimension at most max(1, dim G-I), and such that i(X v Y) = i(X) v i(Y) whenever X, YE G and X v Y # 1. (For example, the inclusion map is an isometry from the 2-dimensional lattice of points and lines of PG(n, K) into PG(n, K), where n 2 2. This definition thus properly contains the corresponding one in [3].)