  Results of R. Stanley and M. Masuda completely characterize the h-vectors of
simplicial posets whose order complexes are spheres. In this paper we examine
the corresponding question in the case where the order complex is a ball. Using
the face rings of these posets, we develop a series of new conditions on their
h-vectors. We also present new methods for constructing poset balls with
specific h-vectors. These results allow us to give a complete characterization
of the h-vectors of simplicial poset balls up through dimension six.
