Positive undecidable numberings in the Ershov hierarchy

A sufficient condition is given under which an infinite computable family of $\Sigma_a^{-1}$-sets has computable positive but undecidable numberings, where $a$ is a notation for a nonzero computable ordinal. This extends a theorem proved by Talasbaeva in [Algebra and Logic, 42, No. 6 (2003), 737–746] for finite levels of the Ershov hierarchy. As a consequence, it is stated that the family of all $\Sigma_a^{-1}$-sets has a computable positive undecidable numbering. In addition, for every ordinal notation $a>1$, an infinite family of $\Sigma_a^{-1}$-sets is constructed which possesses a computable positive numbering but has no computable Friedberg numberings. This answers the question of whether such families exist at any – finite or infinite – level of the Ershov hierarchy, which was originally raised by Badaev and Goncharov only for the finite levels bigger than 1.