# {\alpha} degrees as an automorphism base for the {\alpha}-enumeration degrees

## TL;DR

This paper generalizes Selman's Theorem from classical Computability Theory to the setting of lpha-Computability Theory, establishing conditions under which lpha-degrees form an automorphism base, extending known results to broader ordinal contexts.

## Contribution

The paper extends Selman's Theorem to lpha-Computability Theory, providing a full generalization for regular cardinals and partial results for admissible ordinals.

## Key findings

- lpha-degrees form an automorphism base under certain conditions
- Full generalization for regular cardinals lpha
- Partial results for admissible ordinals lpha

## Abstract

Selman's Theorem in classical Computability Theory gives a characterization of the enumeration reducibility for arbitrary sets in terms of the enumeration reducibility on the total sets: $A \le_e B \iff \forall X [X \equiv_{e} X \oplus \overline{X} \land B \le_{e} X \oplus \overline{X} \implies A \le_{e} X \oplus. \overline{X} ]$. This theorem implies directly that the Turing degrees are an automorphism base of the enumeration degrees. We lift the classical proof to the setting of the $\alpha$-Computability Theory to obtain the full generalization when $\alpha$ is a regular cardinal and partial results for a general admissible ordinal $\alpha$.