Towards characterizing equality in correlation inequalities
type: Item
instance of: scholarly article
title: Towards characterizing equality in correlation inequalities
zbMATH Open document ID: 0836.06007
DOI: 10.1016/0195-6698(95)90015-2
author: Levon H. Khachatrian
author: Rudolf Ahlswede
published in: European Journal of Combinatorics
publication date: 1996-05-02T00:00:00Z
review text: For most of the basic inequalities in mathematics we know conditions which completely specify the cases of equality. This is still not the case for the AD-inequality, which contains many known correlation inequalities as special cases. For some of them the equality characterisation problem could be solved [\textit{R. Aharoni} and \textit{R. Holzman}, J. Lond. Math. Soc., II. Ser. 48, No. 3, 385-395 (1993; Zbl 0799.04010); \textit{I. Beck}, J. Comb. Theory, Ser. A 54, No. 1, 123-128 (1990; Zbl 0699.06002)]. Continuing work of \textit{D. E. Daykin}, \textit{D. J. Kleitman} and \textit{D. B. West} [J. Comb. Theory, Ser. A 26, 135-156 (1979; Zbl 0421.06004)] for the inequality \(|A ||B |< |L ||A \wedge B |\), where the lattice \(L\) is a product of chains of varying lengths and \(A \wedge B = \{a \wedge b : a \in A,\;b \in B\}\), we completely characterize equality in terms of cardinalities of \(A\), \(B\) (Theorem 2) and for ideals \(A\), \(B\) (Theorem 1). Furthermore, we correct an error in Theorem 6 of Daykin, Kleitman and West's paper [loc. cit.]. Finally, as perhaps our most significant contribution, we extend work of \textit{A. J. Hilton} [Q. J. Math., Oxford, II. Ser. 27, 33- 36 (1976; Zbl 0324.05004)] by solving a problem of Daykin, Kleitman and West's [loc. cit.]\ concerning the total cardinality of a cloud-antichain of length 2 with a polarity constraint. Unfortunately the historical remarks of the paper are from an erroneous source.
Mathematics Subject Classification ID: 06D05
Mathematics Subject Classification ID: 05A20
Mathematics Subject Classification ID: 05A19
Mathematics Subject Classification ID: 05D05
zbMATH DE Number: 790739
zbMATH Keywords: correlation inequalities
zbMATH Keywords: equality characterisation
zbMATH Keywords: cloud-antichain
zbMATH Keywords: polarity
MaRDI profile type: Publication
full work available at URL: 0195-6698(95)90015-2
OpenAlex ID: W2006797898
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description: scientific article; zbMATH DE number 790739