Positive presentations of families relative to $e$-oracles

We introduce the notion of $A$-numbering which generalizes the classical notion of numbering. All main attributes of classical numberings are carried over to the objects considered here. The problem is investigated of the existence of positive and decidable computable $A$-numberings for the natural families of sets $e$-reducible to a fixed set. We prove that, for every computable $A$-family containing an inclusion-greatest set, there also exists a positive computable $A$-numbering. Furthermore, for certain families we construct a decidable (and even single-valued) computable total $A$-numbering when $A$ is a low set; we also consider a relativization containing all cases of total sets (this in fact corresponds to computability with a usual oracle).