Absolute and Relative Properties of Negatively Numbered Families

We study in this paper negative A-numberings where A are admissible structures. We establish that a series of classical assumptions is remained to hold for negative A-numberings in the case of the application of certain limits as for numberings as for admissible structures. We find examples of admissible structures A whose families of all Σ-subsets have negative non-decidable minimal computable A-numberings. The admissible structures from this series have negative computable A-numberings whose numbered equivalence differs from corresponding ones of any computable A-numbering of family of total functions.