Computable isomorphism

In computability theory two sets $$A, B$$ of natural numbers are computably isomorphic or recursively isomorphic if there exists a total computable and bijective function $$f \colon \N \to \N$$ such that the image of $$f$$ restricted to $$A\subseteq \N$$ equals $$B\subseteq \N$$, i.e. $$f(A) = B$$.

Further, two numberings $$\nu$$ and $$\mu$$ are called computably isomorphic if there exists a computable bijection $$f$$ so that $$\nu = \mu \circ f$$. Computably isomorphic numberings induce the same notion of computability on a set.

Theorems
By the Myhill isomorphism theorem, the relation of computable isomorphism coincides with the relation of mutual one-one reducibility.