Cylindric numbering

In computability theory a cylindric numbering is a special kind of numbering first introduced by Yuri L. Ershov in 1973.

If a numbering $$\nu$$ is reducible to $$\mu$$ then there exists a computable function $$f$$ with $$\nu = \mu \circ f$$. Usually $$f$$ is not injective, but if $$\mu$$ is a cylindric numbering we can always find an injective $$f$$.

Definition
A numbering $$\nu$$ is called cylindric if
 * $$\nu \equiv_1 c(\nu).$$

That is if it is one-equivalent to its cylindrification

A set $$S$$ is called cylindric if its indicator function
 * $$1_S: \mathbb{N} \to \{0,1\}$$

is a cylindric numbering.

Examples

 * Every Gödel numbering is cylindric

Properties

 * Cylindric numberings are idempotent: $$\nu \circ \nu = \nu$$