Complete numerations with infinitely many singular elements

Denisov, S. D.; Lavrov, I. A.

doi:10.1007/bf02321892

Cited by 5 publications

(4 citation statements)

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“…The study of the sets of special elements of complete numberings was initiated by Denisov and Lavrov in their paper [9]. From results by Khisamiev [11], it follows that every Σ 0 n -computable family containing the least element under inclusion has a Σ 0 ncomputable numbering complete simultaneously with respect to all of its elements.…”

Section: Complete Non-principal Numberingsmentioning

confidence: 99%

“…Another key property of the Gödel numbering is that for any partially computable function ψ there exists a computable function f such that, for every x, W f (x) = W ψ(x) if ψ(x) converges, and W f (x) = ∅ otherwise. This property called by Mal'tsev [5,6] the completeness (with respect to ∅) is also actively studied in the theory of numberings (cf., e.g., [1,[7][8][9][10][11][12][13]) and was used by Ershov [14] to prove Kleene's recursion theorems in arbitrary (not necessarily computable) numberings (i.e. surjective mappings from N onto nonempty countable sets).…”

Section: Introductionmentioning

confidence: 99%

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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: Complete Non-principal Numberingsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

On Non-principal Arithmetical Numberings and Families

Faizrahmanov

2024

Theory Comput Syst

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“…Products of such numberings and numberings with one special element yield numberings with a finite number k of special elements and with a finite number of non-special elements, multiple to the k, or with an infinite number of non-special elements. In [1,4] are examples of the numberings with infinitely many special elements. From [4, Thm.…”

Section: Preliminariesmentioning

confidence: 99%

Subfamilies of special elements of complete numberings

Khisamiev¹

2006

Algebr Logic

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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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Complete numerations with infinitely many singular elements

Cited by 5 publications

References 1 publication

On Non-principal Arithmetical Numberings and Families

On Non-principal Arithmetical Numberings and Families

Subfamilies of special elements of complete numberings

On non-principal arithmetical numberings and families

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