Khutoretskii’s Theorem for Generalized Computable Families

Faizrakhmanov, M. Kh.

doi:10.1007/s10469-019-09557-9

Cited by 7 publications

(2 citation statements)

References 7 publications

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“…Another motivation for studying non-principal Σ 0 n -families is that some structural properties of their numberings are proved for principal and non-principal families separately (cf., e.g., [16][17][18]). In Sections 4 and 5, we prove that for every Σ 0 n -computable numbering, say ν, of a non-principal Σ 0 n -computable family there exists its minimal cover (for arbitrary Σ 0 n -computable families this question was raised by Badaev and Podzorov in their paper [19]) and, if ν is not the least, a Σ 0 n -computable numbering that is incomparable with ν.…”

Section: Introductionmentioning

confidence: 99%

On Non-principal Arithmetical Numberings and Families

Faizrahmanov

2024

Theory Comput Syst

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The paper studies Σ 0 n -computable families (n ⩾ 2) and their numberings. It is proved that any non-trivial Σ 0 n -computable family has a complete with respect to any of its elements Σ 0 n -computable non-principal numbering. It is established that if a Σ 0 n -computable family is not principal, then any of its Σ 0 n -computable numberings has a minimal cover and, if the family is infinite, is incomparable with one of its minimal Σ 0 n -computable numberings. It is also shown that for any Σ 0 ncomputable numbering ν of a Σ 0 n -computable non-principal family there exists its Σ 0 n -computable numbering that is incomparable with ν. If a non-trivial Σ 0 ncomputable family contains the least and greatest elements under inclusion, then for any of its Σ 0 n -computable non-principal non-least numberings ν there exists a Σ 0 n -computable numbering of the family incomparable with ν. In particular, this is true for the family of all Σ 0 n -sets and for the families consisting of two inclusion-comparable Σ 0 n -sets (semilattices of the Σ 0 n -computable numberings of such families are isomorphic to the semilattice of m-degrees of Σ 0 n -sets).

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Section: Introductionmentioning

confidence: 99%

On Non-principal Arithmetical Numberings and Families

Faizrahmanov

2024

Theory Comput Syst

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Section: Introductionmentioning

confidence: 99%

On non-principal arithmetical numberings and families

Faizrahmanov

2023

Preprint

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The paper studies $\Sigma^0_n$ -computable families ( $n\geqslant 2$ ) and their numberings. It is proved that any non-trivial $\Sigma^0_n$-computable family has a complete with respect to any of its elements $\Sigma^0_n$-computable non-principal numbering. It is established that if a $\Sigma^0_n$-computable family is not principal, then any of its $\Sigma^0_n$-computable numberings has a minimal cover and, if the family is infinite, is incomparable with one of its minimal $\Sigma^0_n$-computable numberings. It is also shown that for any $\Sigma^0_n$-computable numbering $\nu$ of a $\Sigma^0_n$-computable non-principal family there exists its $\Sigma^0_n$-computable numbering that is incomparable with $\nu$. If a non-trivial $\Sigma^0_n$-computable family contains the least and greatest elements under inclusion,then for any of its $\Sigma^0_n$-computable non-principal non-least numberings $\nu$ there exists a $\Sigma^0_n$-computable numbering of the family incomparable with $\nu$. In particular, this is true for the family of all $\Sigma^0_n$-sets and for the families consisting of two inclusion-comparable $\Sigma^0_n$-sets (semilattices of the $\Sigma^0_n$-computable numberings of such families are isomorphic to the semilattice of $m$ -degrees of $\Sigma^0_n$-sets). MSC Classification: 03D45

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Some properties of precompletely and positively numbered sets

Faizrahmanov

2025

Annals of Pure and Applied Logic

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Khutoretskii’s Theorem for Generalized Computable Families

Cited by 7 publications

References 7 publications

On Non-principal Arithmetical Numberings and Families

On Non-principal Arithmetical Numberings and Families

On non-principal arithmetical numberings and families

Some properties of precompletely and positively numbered sets

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