  We prove a theorem allowing us to find convex-ear decompositions for
rank-selected subposets of posets that are unions of Boolean sublattices in a
coherent fashion. We then apply this theorem to geometric lattices and face
posets of shellable complexes, obtaining new inequalities for their h-vectors.
Finally, we use the latter decomposition to prove new inequalities for the flag
h-vectors of face posets of Cohen-Macaulay complexes.
