Left-modular elements

Left-modularity is a concept that generalizes 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 Partial Factorization Theorem for a geometric lattice with a modular element. Both formulae provide us with inductive proofs of Blass and Sagan's Total Factorization Theorem for LL lattices. The characteristic polynomials and Mobius functions of non-crossing partition lattices and shuffle posets are computed as examples.