On the relation of $\Sigma$-reducibility between admissible sets

Reducibility on admissible sets is studied which is a stronger version of the usual $\Sigma$-presentability of models. One of its informal prototypes is the interpretability of one computational device in the other. We obtain criteria of reducibility for recursively listed and pure sets, introduce the notion of jump, and prove exact boundaries for the ordinals of jumps. We also show that this reducibility is lifted to $\mathbb{HYP}$-superstructures. Several results are proven on the relations of this reducibility to some known reducibilities.