Equality in the Ahlswede–Daykin and FKG inequalities holds if and only if the underlying lattice decomposes as a direct product and the functions cross-factor across the two components.
Marica and Johanan Sch\"onheim, Differences of sets and a problem of Graham, Canad.\ Math.\ Bull
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Equality conditions for correlation inequalities
Equality in the Ahlswede–Daykin and FKG inequalities holds if and only if the underlying lattice decomposes as a direct product and the functions cross-factor across the two components.