  In this paper we give convex-ear decompositions for the order complexes of
several classes of posets, namely supersolvable lattices with non-zero Mobius
functions and rank-selected subposets of such lattices, rank-selected geometric
lattices, and rank-selected face posets of shellable complexes which do not
include the top rank. These decompositions give us many new inequalities for
the h-vectors of these complexes. In addition, our decomposition of
rank-selected face posets of shellable complexes allows us to prove
inequalities for the flag h-vector of face posets of Cohen-Macaulay complexes.
