Generalized Polymorphisms

05/17/2023

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by   Gilad Chase, et al.

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We determine all m-ary Boolean functions f_0,…,f_m and n-ary Boolean functions g_0,…,g_n satisfying the equation f_0(g_1(z_11,…,z_1m),…,g_n(z_n1,…,z_nm)) = g_0(f_1(z_11,…,z_n1),…,f_m(z_1m,…,z_nm)), for all Boolean inputs { z_ij : i ∈ [n], j ∈ [m] }. This extends characterizations by Dokow and Holzman (who considered the case g_0 = ⋯ = g_n) and by Chase, Filmus, Minzer, Mossel and Saurabh (who considered the case g_1 = ⋯ = g_n).