Quotient posets and the characteristic polynomial

In this dissertation we consider the generating function for the Mobius function of finite partially ordered sets. This generating function is called the characteristic polynomial of the partially ordered set. We are primarily interested in explaining why certain families of partially ordered sets have characteristic polynomials where all the roots are nonnegative integers. To this end we introduce the concept of a homogeneous quotient. These quotients allow us to give new proofs of some well-known results in the literature as well as give generalizations of them. We finish by showing how to use homogeneous quotients to give unified proofs of some classic results about the Mobius function.