A Generalized Macaulay Theorem and Generalized Face Rings

Details A Generalized Macaulay Theorem and Generalized Face Rings A Generalized Macaulay Theorem and Generalized Face Rings

2005-05-16

arxiv.org/abs/math/0505330v2

JCTA 113 (2006) 1321-1331

Mathematics

Combinatorics

Final version: 13 pages, 2 figures. Improved presentation, more detailed proofs, same results. To appear in JCTA

Scientific paper

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.

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