Descriptive properties on admissible sets

Relations are treated between the following descriptive properties on admissible sets: enumerability, uniformization, reduction, separation, extension. Moreover, in the setting of these properties, we consider existence problems for a universal computable function and for a computable function universal for $\{0;1\}$-valued computable functions. It is shown that all relations between the given properties are strict. Also we look into algorithmic complexity of admissible sets lending support to the specified relations. It is stated that the reduction principle fails in some admissible sets over classical structures.