On the resemblance and recursive isomorphism types of partial recursive functions (Q1975814)

Two partial computable functions \(\alpha\) and \(\beta\) are called isomorphic whenever there exists a computable permutation \(f\) with \(\alpha=f^{-1}\beta f\); the functions are called resembling whenever there exist computable permutations \(f\) and \(g\) with \(\alpha=f^{-1}\beta g\). A partial computable nonempty function is called a \(t\)-function whenever it is not constant and its resemblance type coincides with its isomorphism type. \textit{E. A. Polyakov} [Sib. Math. Zh. 30, No. 6, 188-192 (1989; Zbl 0714.03034)] proved that, if the domain of a \(t\)-function \(\alpha\) is not simple, then its range is the whole \(N\). The author proves this result for \(t\)-functions with simple domains and proves that all the level sets \(\{ x\mid \alpha(x)=a\}\) for \(t\)-functions are recursively isomorphic.