TY - JOUR
T1 - A generalized Macaulay theorem and generalized face rings
AU - Nevo, Eran
N1 - Funding Information:
I thank my advisor Prof. Gil Kalai for many helpful discussions, and Prof. Anders Björner for his comments on earlier versions of this paper. Part of this work was done during the author’s stay at Institut Mittag-Leffler, supported by the ACE network. I thank the referees, especially one of them, for their suggestions, which improved the presentation in this paper.
PY - 2006/10/1
Y1 - 2006/10/1
N2 - 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.
AB - 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.
KW - Face ring
KW - Kruskal-Katona
KW - Macaulay inequalities
KW - Meet semi-lattice
KW - Shadow
UR - http://www.scopus.com/inward/record.url?scp=33746640559&partnerID=8YFLogxK
U2 - 10.1016/j.jcta.2005.12.002
DO - 10.1016/j.jcta.2005.12.002
M3 - Article
AN - SCOPUS:33746640559
SN - 0097-3165
VL - 113
SP - 1321
EP - 1331
JO - Journal of Combinatorial Theory. Series A
JF - Journal of Combinatorial Theory. Series A
IS - 7
ER -