Left-Modular Elements of Lattices

Left-modularity is a concept that generalizes the notion of modularity in lattice theory. In this paper, we give a characterization of left-modular elements and derive two formulae for the characteristic polynomial, ?, of a lattice with such an element, one of which generalizes Stanley's theorem 6] about the partial factorization of ? in a geometric lattice. Both formulae provide us with inductive proofs for Blass and Sagan's theorem 2] about the total factorization of ? in LL lattices. The characteristic polynomials and the Mobius functions of non-crossing partition lattices and shuffle posets are computed as examples.