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 [G. Wegner, Kruskal–Katona's theorem in generalized complexes, in: Finite and Infinite Sets, vol. 2, in: Colloq. Math. Soc. János Bolyai, vol. 37, North-Holland, Amsterdam, 1984, pp. 821–828], as special cases. Specializing the proof to the later family, one obtains the Kruskal–Katona inequalities and their proof as in [G. Wegner, Kruskal–Katona's theorem in generalized complexes, in: Finite and Infinite Sets, vol. 2, in: Colloq. Math. Soc. János Bolyai, vol. 37, North-Holland, Amsterdam, 1984, pp. 821–828].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.