Some modifications of iterated Kleene calculability (Q1097877)

\textit{N. V. Belyakin} [Mat. Sbornik, Nov. Ser. 101(143), 21-43 (1976; Zbl 0342.02031); Algebra Logika 22, No.1, 3-25 (1983; Zbl 0538.03040)] considered a mechanism for constructing recursive hierarchies, iterated Kleene computability (IKC); along an ordinal enumeration \(\nu\) we generate oracles of increasing strength, Kleene relativized to a functional G of type 2, i.e., these oracles 1) calculate the functional G; 2) uniformly in the \(\nu\)-indices, solve the graphs of the preceding oracles (answer the ``problems of the past''). In this article we consider variations of conditions 1 and 2; in Sec. 2 we study a method of iterating an arbitrary monotonic or extending operator; in Sec. 3 we consider iterated Kleene computability with respect to multivalued functionals of type 2, simultaneously considered for three weaker variants of condition 2. We show that for almost all variations of condition 2, we do not obtain a great difference from IKC, whereas changing condition 1 can have a great effect on the properties of hierarchies.