Geodesic
Minimal generalized computable numberings and families of positive preorders
Algebra i logika, Tome 61 (2022) no. 3, pp. 280-307

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We study $A$-computable numberings for various natural classes of sets. For an arbitrary oracle $A\geq_T \mathbf{0'}$, an example of an $A$-computable family $S$ is constructed in which each $A$-computable numbering of $S$ has a minimal cover, and at the same time, $S$ does not satisfy the sufficient conditions for the existence of minimal covers specified by S. A. Badaev and S. Yu. Podzorov in [Sib. Math. J., 43, No. 4, 616–622 (2002)]. It is proved that the family of all positive linear preorders has an $A$-computable numbering iff $A' \geq_T \mathbf{0}''$. We obtain a series of results on minimal $A$-computable numberings, in particular, Friedberg numberings and positive undecidable numberings.
Keywords: $A$-computable numbering, positive linear preorder, Rogers semilattice, Friedberg numbering, positive numbering, minimal cover. 
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     author = {F. Rakymzhankyzy and N. A. Bazhenov and A. A. Issakhov and B. S. Kalmurzayev},
     title = {Minimal generalized computable numberings and families of positive preorders},
     journal = {Algebra i logika},
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     year = {2022},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/AL_2022_61_3_a1/}
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F. Rakymzhankyzy; N. A. Bazhenov; A. A. Issakhov; B. S. Kalmurzayev. Minimal generalized computable numberings and families of positive preorders. Algebra i logika, Tome 61 (2022) no. 3, pp. 280-307. http://geodesic.mathdoc.fr/item/AL_2022_61_3_a1/