# Kalimullin Pair and Semicomputability in $\alpha$-Computability Theory

## TL;DR

This paper extends semicomputability concepts to $oldsymbol{ extalpha}$-Computability Theory, introducing $oldsymbol{ extalpha}$-Kalimullin pairs and analyzing their definability and structural properties in $oldsymbol{ extalpha}$-enumeration degrees.

## Contribution

It defines $oldsymbol{ extalpha}$-Kalimullin pairs, proves their definability under certain conditions, and shows that every nontrivial total $oldsymbol{ extalpha}$-enumeration degree is a join of a maximal $oldsymbol{ extalpha}$-Kalimullin pair.

## Key findings

- $oldsymbol{ extalpha}$-Kalimullin pairs are definable in $oldsymbol{ extalpha}$-enumeration degrees for certain $oldsymbol{ extalpha}$.
- Every nontrivial total $oldsymbol{ extalpha}$-enumeration degree is a join of a maximal $oldsymbol{ extalpha}$-Kalimullin pair when $oldsymbol{ extalpha}$ is an infinite regular cardinal.
- The results generalize classical semicomputability to the setting of $oldsymbol{ extalpha}$-computability theory.

## Abstract

We generalize some results on semicomputability by Jockusch \cite{jockusch1968semirecursive} to the setting of $\alpha$-Computability Theory. We define an $\alpha$-Kalimullin pair and show that it is definable in the $\alpha$-enumeration degrees $\mathcal{D}_{\alpha e}$ if the projectum of $\alpha$ is $\alpha^*=\omega$ or if $\alpha$ is an infinite regular cardinal. Finally using this work on $\alpha$-semicomputability and $\alpha$-Kalimullin pairs we conclude that every nontrivial total $\alpha$-enumeration degree is a join of a maximal $\alpha$-Kalimullin pair if $\alpha$ is an infinite regular cardinal.