A conditions for uniqueness reresentation of p-Logic function into disjunctive product of functions

Let f : V- Z_p be p-logic function, n > 2, and V- = Zp is considered as a vector space over Zp. A disjunctive decomposition of f into a product of p-logic functions under various linear transformations of arguments is considered. Function f is linearly decomposable into disjunctive product if there exists a linear transformation A of the vector space Vn such that f(xA) = f1(x1, . . . , xk ) f2 (xk+1 , . . . , xn) for some k, 1 < k < n, and functions f₁ and f₂. We say that argument x_n of functions f(x) is essential iff f(x) = f(x + en) for en = (0, . . . , 0, 1). The main result is: if all arguments of all functions f(xA) under linear substitutuions A of the vector space Vn are essential, the set {a G V_n : f (a) = 0} is not contained in hyperplane of Vn, and f is linearly decompsable into the disjunctive product f₁.....f_m, where m is maximal, then the direct sum of subspaces V_n = V ⁽¹⁾ + . . . + V ^(m) is unique and invariant under the stabilizer group of the function f in general linear group.

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