Abstract This paper studies two approaches to relativizing the notions of complete and precomplete numbering. The first one was introduced by Selivanov in the late 1980s, and it strengthens the standard definitions of complete and precomplete numbering. The second one was introduced by Badaev, Goncharov, and Sorbi in the early 2000s, and it is the full relativization of these two concepts. In the first part of the paper, we study how these two approaches differ from each other. In the second part, we study Mal’cev’s object uniquely for the relativized complete numberings.

Abstract This paper considers various forms of the fixed point theorem (Kleene’s recursion theorem with and without parameters, Arslanov’s and Ershov’s recursion theorems) in arbitrary numberings. We prove that the classes of minimal numberings satisfying different of the listed forms of the fixed point theorem are pairwise distinct.

In the article, we study the behaviour of enumeration jumps of sets of low e-degrees in the Ershov hierarchy.