Left-modular elements

TL;DR

This paper characterizes left-modular elements in lattice theory, derives formulas for their characteristic polynomials, and applies these results to non-crossing partition lattices and shuffle posets.

Contribution

It introduces a new characterization of left-modular elements and generalizes Stanley's Partial Factorization Theorem for geometric lattices.

Findings

Derived two formulas for characteristic polynomials involving left-modular elements

Provided inductive proofs of Blass and Sagan's Total Factorization Theorem for LL lattices

Computed characteristic polynomials and Mobius functions for specific lattices

Abstract

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.