Geodesic
Some absolute properties of $A$-computable numberings
Algebra i logika, Tome 57 (2018) no. 4, pp. 426-447.

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For an arbitrary set $A$ of natural numbers, we prove the following statements: every finite family of $A$-computable sets containing a least element under inclusion has an $A$-computable universal numbering; every infinite $A$-computable family of total functions has (up to $A$-equivalence) either one $A$-computable Friedberg numbering or infinitely many such numberings; every $A$-computable family of total functions which contains a limit function has no $A$-computable universal numberings, even with respect to $A$-reducibility.
Keywords: $A$-computable numbering, $A$-computable Friedberg numbering, $A$-computable universal numbering, $A$-reducibility. 
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S. A. Badaev; A. A. Issakhov. Some absolute properties of $A$-computable numberings. Algebra i logika, Tome 57 (2018) no. 4, pp. 426-447. http://geodesic.mathdoc.fr/item/AL_2018_57_4_a1/

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