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, χ, of a lattice with such an element, one of which generalizes Stanleyʹs theorem [6] about the partial factorization of χ in a geometric lattice. Both formulae provide us with inductive proofs for Blass and Saganʹs theorem [2] about the total factorization of χ in LL lattices. The characteristic polynomials and the Möbius functions of non-crossing partition lattices and shuffle posets are computed as examples.