  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.
