Generating sets for lattices of dimension two

The dimension of a partially ordered set P is the smallest integer n (if it exists) such that the partial order on P is the intersection of n linear orders. It is shown that if L is a lattice of dimension two containing a sublattice isomorphic to the modular lattice M_2n+1, then every generating set of L has at least n+2 elements. A consequence is that every finitely generated lattice of dimension two and with no infinite chains is finite.