Article collection "Mathematical Problems of Cybernetics" №19, Moscow, 2019
Multivalued logic classes closed with respect to strengthened superposition operation
The paper studies k-valued logic functions for k = 2m, m ≥ 2. Such functions are encoded in the binary number system and a special β-closure operator is deﬁned. The completeness criteria for β-closed classes is obtained and all maximal classes are described for this operator. It is proven that the set of β-closed classes of functions, taking no more than two values, is countably inﬁnite and the description of such classes is provided. A closed class of Boolean functions is associated with every set of the k-valued logic functions and called its boolean closure. For every r = 3, 4 and every Boolean closed class B it is shown that the family of β-closed classes with boolean closure equal to B and containing only functions, taking not more than r values, is ﬁnite or continuous. For given r the explicit partition of all Boolean closed classes with respect to this criteria is provided.