Towards characterizing equality in correlation inequalities (Q1896333)

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.