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Generalization of two Belousov’s theorems for strongly dependent functions of k-valued logic. (Russian) Zbl 0591.03011

A representation of functions of two-valued logic by means of irreversible superpositions of functions of k-valued logic falls into the category of interesting problems of k-valued logic [A. V. Kuznetsov, Tr. Mat. Inst. Steklova 51, 186-225 (1958; Zbl 0092.25304)]. A concept of strongly dependent function was introduced for the development of such representation [L. M. Sosinskij, Probl. Kibern. 12, 57-68 (1964)]. A function \(f(x_ 0,x_ 1,...,x_ n)\) is strongly dependent if for any variable \(x_ i\) there is an n-dimensional set of elements \((a_ 0,a_ 1,...,a_{i-1},a_{i+1},...,a_ n)\in Q^ n\) for which the function \(f(a_ 0,a_ 1,...,a_{i-1},x_ i,a_{i+1},...,a_ n)\) is a substitution on a set Q (here Q is the domain of definition and the value range of the function \(f(x_ 0,x_ 1,...,x_ n))\). The set of strongly dependent functions with the irreversible superpositions forms a positional algebra [V. D. Belousov, n-ary quasi-groups (Russian) (1972; Zbl 0282.20061)].
The main result of the present paper consists in a more compact and transparent proof of Belousov’s theorems connected with the solution of special-type equations in a positional algebra. A number of results for quasi-groups generalizing the investigations of Belousov [loc. cit.] is obtained.
Reviewer: G.E.Tseytlin

MSC:

03B50 Many-valued logic
20N05 Loops, quasigroups
03G25 Other algebras related to logic
94C10 Switching theory, application of Boolean algebra; Boolean functions (MSC2010)
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