Geodesic
Negative numberings in admissible sets.~I
Matematičeskie trudy, Tome 26 (2023) no. 1, pp. 47-92

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We construct an admissible set $\mathbb{A}$ such that the family of all $\mathbb{A}$-computably enumerable sets possesses a negative computable $\mathbb{A}$-numbering but lacks positive computable $\mathbb{A}$-numberings. We also discuss the question on existence of minimal negative $\mathbb{A}$-numberings. 
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     author = {I. Sh. Kalimullin and V. G. Puzarenko and M. Kh. Faizrahmanov},
     title = {Negative numberings in admissible {sets.~I}},
     journal = {Matemati\v{c}eskie trudy},
     pages = {47--92},
     publisher = {mathdoc},
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     number = {1},
     year = {2023},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MT_2023_26_1_a3/}
}
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I. Sh. Kalimullin; V. G. Puzarenko; M. Kh. Faizrahmanov. Negative numberings in admissible sets.~I. Matematičeskie trudy, Tome 26 (2023) no. 1, pp. 47-92. http://geodesic.mathdoc.fr/item/MT_2023_26_1_a3/