On properties of multiaffine predicates on a finite set

We consider predicates on a finite set that are invariant with respect to an affine operation $f_G$, where $G$ is some Abelian group. Such predicates are said to be multiaffine for the group $G$. Special attention is paid to predicates that are affine for a group $G_q$ of addition modulo $q = p^s$, where $p$ is a prime number and $s \geqslant 1$. We establish the predicate multiaffinity criterion for a group $G_q$. Then we introduce disjunctive normal forms (DNF) for predicates on a finite set and obtain properties of DNFs of predicates that are multiaffine for a group $G_q$. Finally we show how these properties can be used to design a polynomial algorithm that decides satisfiability of a system of predicates which are multiaffine for a group $G_q$, if predicates are specified by DNF.