Abstract
In the paper, generalizations of principal and complete numberings are studied, the so-called \(e\) -principal and \(e\) -complete numberings, respectively, that are consistent with the \(e\) -reducibility of numberings introduced by Degtev. It is proven that, for an arbitrary set \(A\) , every finite family of \(A\) -computably enumerable sets has an \(A\) -computable \(e\) -principal numbering. Necessary and sufficient conditions are obtained for the Turing completeness of the set \(A\) in terms of \(e\) -principal and \(e\) -complete numberings of \(A\) -computable families. It is established that the classes of \(e\) -complete and precomplete numberings are not comparable with respect to inclusion, and also, for every Turing complete set \(A\) and every infinite \(A\) -computable family, its \(e\) -complete \(A\) -computable numbering is constructed, which is both \(e\) -minimal and minimal.
