  We prove that the $f$-vector of members in a certain class of meet
semi-lattices satisfies Macaulay inequalities. We construct a large family of
meet semi-lattices belonging to this class, which includes all posets of
multicomplexes, as well as meet semi-lattices with the "diamond property",
discussed by Wegner, as spacial cases. Specializing the proof to that later
family, one obtains the Kruskal-Katona inequalities and their proof as in
Wegner's.
  For geometric meet semi lattices we construct an analogue of the exterior
face ring, generalizing the classic construction for simplicial complexes. For
a more general class, which include also multicomplexes, we construct an
analogue of the Stanley-Reisner ring. These two constructions provide algebraic
counterparts (and thus also algebraic proofs) of Kruskal-Katona's and
Macaulay's inequalities for these classes, respectively.
