The aim of this paper is to present a general approach to designing efficient algorithms intended for checking some properties (monotonicity, some specific variants of precompleteness, etc.) of multiple-valued functions represented by polynomials. The properties under consideration are characterized by predicates. The key idea of this approach is based upon the extension of the concept of transitivity to predicates of arbitrary arity. We demonstrate that whenever multiple-valued functions are represented by polynomials and some set of functions is characterized by an extended transitive and total reflexive predicate, then the membership problem for this class is decidable in polynomial time.
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