A rank inequality for finite geometric lattices

Abstract

Let L be a finite geometric lattice of dimension n, and let w(k) denote the number of elements in L of rank k. Two theorems about the numbers w(k) are proved: first, w(k)≥w(1) for k=2,3,…,n−1. Second, w(k)=w(1) if and only if k=n−1 and L is modular. Several corollaries concerning the “matching” of points and dual points are derived from these results.