  In this paper we consider the characteristic polynomial 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 the
Tamari 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. Finally we will use quotient posets to give unified
proofs of some classic M\"obius function results.
