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Isomorphism Types of Rogers Semilattices in the Analytical Hierarchy. / Bazhenov, Nikolay; Ospichev, Sergey; Yamaleev, Mars.

Lecture Notes Series, Institute for Mathematical Sciences. World Scientific, 2024. p. 97-114 (Lecture Notes Series, Institute for Mathematical Sciences; Vol. 42).

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Harvard

Bazhenov, N, Ospichev, S & Yamaleev, M 2024, Isomorphism Types of Rogers Semilattices in the Analytical Hierarchy. in Lecture Notes Series, Institute for Mathematical Sciences. Lecture Notes Series, Institute for Mathematical Sciences, vol. 42, World Scientific, pp. 97-114. https://doi.org/10.1142/9789811278631_0004

APA

Bazhenov, N., Ospichev, S., & Yamaleev, M. (2024). Isomorphism Types of Rogers Semilattices in the Analytical Hierarchy. In Lecture Notes Series, Institute for Mathematical Sciences (pp. 97-114). (Lecture Notes Series, Institute for Mathematical Sciences; Vol. 42). World Scientific. https://doi.org/10.1142/9789811278631_0004

Vancouver

Bazhenov N, Ospichev S, Yamaleev M. Isomorphism Types of Rogers Semilattices in the Analytical Hierarchy. In Lecture Notes Series, Institute for Mathematical Sciences. World Scientific. 2024. p. 97-114. (Lecture Notes Series, Institute for Mathematical Sciences). doi: 10.1142/9789811278631_0004

Author

Bazhenov, Nikolay ; Ospichev, Sergey ; Yamaleev, Mars. / Isomorphism Types of Rogers Semilattices in the Analytical Hierarchy. Lecture Notes Series, Institute for Mathematical Sciences. World Scientific, 2024. pp. 97-114 (Lecture Notes Series, Institute for Mathematical Sciences).

BibTeX

@inbook{5636df2c8fcd45d782d5782af53885e1,
title = "Isomorphism Types of Rogers Semilattices in the Analytical Hierarchy",
abstract = "A numbering of a countable family S is a surjective map from the set of natural numbers ω onto S. A numbering ν is reducible to a numbering µ if there is an effective procedure which given a ν-index of an object from S, computes a µ-index for the same object. The reducibility between numberings gives rise to a class of upper semilattices, which are usually called Rogers semilattices. This chapter studies Rogers semilattices for families S ⊂ P(ω) belonging to various levels of the analytical hierarchy. We prove that for any non-zero natural numbers m ≠ n, any non-trivial Rogers semilattice of a Π1m-computable family cannot be isomorphic to a Rogers semilattice of a Π1n-computable family. One of the key ingredients of the proof is an application of the result by Downey and Knight on degree spectra of linear orders.",
author = "Nikolay Bazhenov and Sergey Ospichev and Mars Yamaleev",
note = "The work of M. Yamaleev was supported by Russian Science Foundation, project No. 18-11-00028. The work of S. Ospichev was funded by RFBR according to the research project No. 17-01-00247. The work of N. Bazhenov was supported by Nazarbayev University Faculty Development Competitive Research Grants N090118FD5342.",
year = "2024",
doi = "10.1142/9789811278631_0004",
language = "English",
isbn = "9811278628",
series = "Lecture Notes Series, Institute for Mathematical Sciences",
publisher = "World Scientific",
pages = "97--114",
booktitle = "Lecture Notes Series, Institute for Mathematical Sciences",
address = "United States",

}

RIS

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T1 - Isomorphism Types of Rogers Semilattices in the Analytical Hierarchy

AU - Bazhenov, Nikolay

AU - Ospichev, Sergey

AU - Yamaleev, Mars

N1 - The work of M. Yamaleev was supported by Russian Science Foundation, project No. 18-11-00028. The work of S. Ospichev was funded by RFBR according to the research project No. 17-01-00247. The work of N. Bazhenov was supported by Nazarbayev University Faculty Development Competitive Research Grants N090118FD5342.

PY - 2024

Y1 - 2024

N2 - A numbering of a countable family S is a surjective map from the set of natural numbers ω onto S. A numbering ν is reducible to a numbering µ if there is an effective procedure which given a ν-index of an object from S, computes a µ-index for the same object. The reducibility between numberings gives rise to a class of upper semilattices, which are usually called Rogers semilattices. This chapter studies Rogers semilattices for families S ⊂ P(ω) belonging to various levels of the analytical hierarchy. We prove that for any non-zero natural numbers m ≠ n, any non-trivial Rogers semilattice of a Π1m-computable family cannot be isomorphic to a Rogers semilattice of a Π1n-computable family. One of the key ingredients of the proof is an application of the result by Downey and Knight on degree spectra of linear orders.

AB - A numbering of a countable family S is a surjective map from the set of natural numbers ω onto S. A numbering ν is reducible to a numbering µ if there is an effective procedure which given a ν-index of an object from S, computes a µ-index for the same object. The reducibility between numberings gives rise to a class of upper semilattices, which are usually called Rogers semilattices. This chapter studies Rogers semilattices for families S ⊂ P(ω) belonging to various levels of the analytical hierarchy. We prove that for any non-zero natural numbers m ≠ n, any non-trivial Rogers semilattice of a Π1m-computable family cannot be isomorphic to a Rogers semilattice of a Π1n-computable family. One of the key ingredients of the proof is an application of the result by Downey and Knight on degree spectra of linear orders.

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M3 - Chapter

SN - 9811278628

T3 - Lecture Notes Series, Institute for Mathematical Sciences

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EP - 114

BT - Lecture Notes Series, Institute for Mathematical Sciences

PB - World Scientific

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