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.
Kleitman, Families of non-disjoint subsets, J.\ Combin.\ Theory 1 (1966), 153--155
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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.