Modular Elements of Geometric Lattices

Let L be a finite geometric lattice of rank n with rank function r. (For definitions, see e.g., [3, Chapter 2], [4], or [1, Chapter 4].) An element x s L is called a modular element if it forms a modular pair with every y e L , i.e., if a<~y then a V ( x A y ) = (a v x )Ay . Recall that in an upper semimodular lattice (and thus in a geometric lattice) the relation of being a modular pair is symmetric; in fact (x, y) is a modular pair i f and only if r (x) + r (y) = r (x v y) + r (x A y) [1, p. 83]. Every point (atom) of a geometric lattice is a modular element. I f every element of L is modular, then L is a modular lattice. The main object of this paper is to show that a modular element of L induces a factorization of the characteristic polynomial of L. This is done in Section 2. First we discuss some other aspects of modular elements. The following theorem provides a characterization of modular elements.