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.
Graham, Applications of the FKG inequality and its relatives, in Mathematical programming: the state of the art, Springer, Berlin, 1983, 115--131
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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.