On resemblance and recursive isomorphism types of partial recursive functions (Q5932630)

Functions \(\alpha\) and \(\beta\) are called resembling if there exist computable permutations \(f\) and \(g\) such that \(\alpha=f\beta g\). We say that these functions are isomorphic if there exists a computable permutation \(f\) such that \(\alpha = f^{-1}\beta f\). The author presents a formula for computing the number \(P(n)\) of recursive isomorphism types within a resemblance type of a partial recursive function whose range consists of \(n\geq 1\) distinct values. The number \(P(n)\) is proven to be a lower bound for the number of recursive isomorphism types within a resemblance type of a partial recursive function whose range contains \(\geq n\) elements and the complement of its domain is infinite. In addition, the author proves that if a partial recursive function \(\alpha\) is neither empty nor constant and its recursive isomorphism type consists of a single resemblance type then \(\alpha\) has no recursive total extensions.