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 xsL is called a modular element if it forms a modular pair with every yeL, i.e., if a<~y then aV(xAy) = (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 modula