  We introduce a new method for showing that the roots of the characteristic
polynomial of certain finite lattices are all nonnegative integers. This method
is based on the notion of a quotient of a poset which will be developed to
explain this factorization. Our main theorem will give two simple conditions
under which the characteristic polynomial factors with nonnegative integer
roots. We will see that Stanley's Supersolvability Theorem is a corollary of
this result. Additionally, we will prove a theorem which gives three conditions
equivalent to factorization. To our knowledge, all other theorems in this area
only give conditions which imply factorization. This theorem will be used to
connect the generating function for increasing spanning forests of a graph to
its chromatic polynomial. We finish by mentioning some other applications of
quotients of posets as well as some open questions.
