Applications of quotient posets

In this paper we consider the characteristic polynomials of not necessarily ranked posets. We do so by allowing the rank to be an arbitrary function from the poset to the nonnegative integers. We will prove two results showing that the characteristic polynomial of a poset has nonnegative integral roots. Our factorization theorems will then be used to show that any interval of a crosscut-simplicial lattice has a characteristic polynomial which factors in this way. Blass and Sagan's result about LL lattices will also be shown to be a consequence of our factorization theorems.