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.
Spencer, The probabilistic method (Fourth ed.), John Wiley, Hoboken, NJ, 2016, 375 pp
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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.