Left-modular elements of lattices

摘要

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, chi, of a lattice with such an element, one of which generalizes Stanley's theorem [6] about the partial factorization of chi in a geometric lattice. Both formulae provide us with inductive proofs for Blass and Sagan's theorem [2] about the total factorization of chi in LL lattices. The characteristic polynomials and the Mobius functions of Don-crossing partition lattices and shuffle posets are computed as examples. (C) 2000 Academic Press AMS 1991 Subject Classifications: Primary 06C10; Secondary 05A15, 06A07. [References: 7]