A generalized Macaulay theorem and generalized face rings

We prove that the f -vector of members in a certain class of meet semi-lattices satisfies Macaulay inequalities 0 ≤ ∂k(fk) ≤ fk−1 for all k ≥ 0. 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 [11], as special cases. Specializing the proof to the later family, one obtains the Kruskal-Katona inequalities and their proof as in [11]. 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 also includes 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.