Superstructures of the class of polynomials in Pk

On the closed classes of multivalued logic containing the polynomial class

“…are distinct primes, r\, ...,r v are natural numbers, v > 2, were closely related to the properties of superstructures of the classes of polynomials in the systems P* ( , where k { -p r . ;, i = 1,..., v. A series of the results obtained along these lines can be found in the papers due to Cherepov [17,18], Meshchaninov [19][20][21][22], and Remizov [23]. In [17], the superstructure of the class 9JI* for square-free k is described (this is precisely the case where 9ft* = Pol*).…”

“…In [18] it was stated that the superstructure of the class 9Jt /; r for r > 2 is isomorphic to the lattice of all subsets of the set {1,..., r -1}. In [21] it was proved that for k = ρ\ρι...ρ ν , where p\,pi, ...,/? v are distinct primes, the class Pol* is precomplete in 97Ϊ*.…”

A method for constructing polynomials of k-valued logic functions

“…These peculiarities allow us to achieve some advantages in applications (the complexity of computations, the possibility of parallel computing) and to prove polynomiality of any /7-periodic function of the class 971 for each composite k (in the case of k = p a polynomial representability of unary and binary /7-periodic functions was established in [7]). On the base of these results the investigation and the complete description of the lattice of the closed classes locating between Pol and 971 were carried out in [14,16]. This lattice was described independently in [18].…”

“…The following general result for arbitrary k was obtained by Kuznetsov [5]: if k is a prime, then each function of fc-valued logic can be represented by a polynomial modulo k\ if k is a composite number, then the polynomials form a proper subclass Pol C P k which is not precomplete in P k . In this connection in [6] the number of η-variable functions of the class Pol for arbitrary n and k was determined, and in [7][8][9][10][11][12][13][14][15][16][17][18][19] the search of conditions for polynomial representability in P k and the description of closed classes containing Pol were started.…”