Cylindrification

In computability theory a cylindrification is a construction that associates a cylindric numbering to each numbering. The concept was first introduced by Yuri L. Ershov in 1973.

Definition
Given a numbering $$\nu$$, the cylindrification $$c(\nu)$$ is defined as
 * $$\mathrm{Domain}(c(\nu)) := \{\langle n, k \rangle | n \in \mathrm{Domain}(\nu)\}$$
 * $$c(\nu)\langle n, k \rangle := \nu(n)$$

where $$\langle n, k \rangle$$ is the Cantor pairing function.

Note that the cylindrification operation increases the input arity by 1.

Properties

 * Given two numberings $$\nu$$ and $$\mu$$ then $$\nu \le \mu \Leftrightarrow c(\nu) \le_1 c(\mu)$$
 * $$\nu \le_1 c(\nu)$$