Kalimullin Pair and Semicomputability in $α$-Computability Theory
Dávid Natingga · arXiv (Cornell University) · 2019
We generalize some results on semicomputability by Jockusch \cite{jockusch1968semirecursive} to the setting of $α$-Computability Theory. We define an $α$-Kalimullin pair and show that it is definable in the $α$-enumeration degrees $\mathcal{D}_{αe}$ if the projectum of $α$ is $α^*=ω$ or if $α$ is an infinite regular cardinal. Finally using this work on $α$-semicomputability and $α$-Kalimullin pairs we conclude that every nontrivial total $α$-enumeration degree is a join of a maximal $α$-Kalimullin pair if $α$ is an infinite regular cardinal.