On the conditions of complete asymptotics of the power of function classes of \(k\)-valued logic that preserve the finitary predicate (Q1965358)

An asymptotic for the logarithm of classes of monotone Boolean functions is well known from the results on Dedekind's problem. The asymptotic for the logarithm of a power for monotone \(k\)-valued functions is known as well. The asymptotic equivalence of the form \[ \log_k|F_{n}|\approx k^n , \tag{1} \] where \(F_{n}\) is a predicate-preserving class of functions for \(n \to \infty\) (\(n\) is the number of fixed variables), is proved in the article. Some necessary and some sufficient conditions for fulfilling the relation (1) are presented.