A certain reducibility on admissible sets

We introduce a certain reducibility on admissible sets which preserves definable predicates. Some lattice-theoretic properties are given of the ordered sets of the classes of admissible sets equivalent under this reducibility. Furthermore, we give a transformation that assigns to each admissible set some hereditarily finite set and preserves the following list of descriptive set-theoretic properties (with account taken of the levels of a definable hierarchy): enumerability, quasiprojectibility, uniformization, existence of a universal function, separation, and extension. We introduce the notion of jump of an admissible set which translates the descriptive set-theoretic properties considered above to the corresponding properties lowering levels by 1.