Generalized Polymorphisms

We determine all $m$-ary Boolean functions $f_0,\ldots,f_m$ and $n$-ary Boolean functions $g_0,\ldots,g_n$ satisfying the equation \[ f_0(g_1(z_{11},\ldots,z_{1m}),\ldots,g_n(z_{n1},\ldots,z_{nm})) = g_0(f_1(z_{11},\ldots,z_{n1}),\ldots,f_m(z_{1m},\ldots,z_{nm})), \] for all Boolean inputs $\{ z_{ij} : i \in [n], j \in [m] \}$. This extends characterizations by Dokow and Holzman (who considered the case $g_0 = \cdots = g_n$) and by Chase, Filmus, Minzer, Mossel and Saurabh (who considered the case $g_1 = \cdots = g_n$).