On the linear disjunctive decomposition of a $p$-logic function into a product of functions

Let $p$ be a prime number, $p\ge 3$. We consider the set of decompositions of a $p$-logic function into a product of functions with disjoint subsets of variables obtained by means of linear substitutions of arguments. Each decomposition of this kind is associated with a decomposition of the vector space into a direct sum of subspaces. We present conditions under which such space decomposition is unique up to rearrangement of subspaces. Also, a criterion for such product to be balanced is given.