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Sibirskii Matematicheskii Zhurnal, 2018, Volume 59, Number 1, Pages 29–40
DOI: https://doi.org/10.17377/smzh.2018.59.103
(Mi smj2951)
 

This article is cited in 10 scientific papers (total in 10 papers)

On dark computably enumerable equivalence relations

N. A. Bazhenovab, B. S. Kalmurzaevc

a Sobolev Institute of Mathematics, Novosibirsk, Russia
b Novosibirsk State University, Novosibirsk, Russia
c Al-Farabi Kazakh National University, Almaty, Kazakhstan
References:
Abstract: We study computably enumerable (c.e.) relations on the set of naturals. A binary relation R on ω is computably reducible to a relation S (which is denoted by RcS) if there exists a computable function f(x) such that the conditions (xRy) and (f(x)Sf(y)) are equivalent for all x and y. An equivalence relation E is called dark if it is incomparable with respect to c with the identity equivalence relation. We prove that, for every dark c.e. equivalence relation E there exists a weakly precomplete dark c.e. relation F such that EcF. As a consequence of this result, we construct an infinite increasing c-chain of weakly precomplete dark c.e. equivalence relations. We also show the existence of a universal c.e. linear order with respect to c.
Keywords: equivalence relation, computably enumerable equivalence relation, computable reducibility, weakly precomplete equivalence relation, computably enumerable order, lo-reducibility.
Funding agency Grant number
Russian Foundation for Basic Research 16-31-60058 мол_а_дк
Ministry of Education and Science of the Republic of Kazakhstan ГФ4/3952
N. A. Bazhenov was supported by the Russian Foundation for Basic Research (Grant 16-31-60058-mol_a_dk). B. S. Kalmurzaev was supported by the Science Committee of the Republic of Kazakhstan (Grant GF4/3952).
Received: 07.06.2017
English version:
Siberian Mathematical Journal, 2018, Volume 59, Issue 1, Pages 22–30
DOI: https://doi.org/10.1134/S0037446618010032
Bibliographic databases:
Document Type: Article
UDC: 510.57
MSC: 35R30
Language: Russian
Citation: N. A. Bazhenov, B. S. Kalmurzaev, “On dark computably enumerable equivalence relations”, Sibirsk. Mat. Zh., 59:1 (2018), 29–40; Siberian Math. J., 59:1 (2018), 22–30
Citation in format AMSBIB
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\paper On dark computably enumerable equivalence relations
\jour Sibirsk. Mat. Zh.
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\vol 59
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\pages 29--40
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\crossref{https://doi.org/10.17377/smzh.2018.59.103}
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\pages 22--30
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  • https://www.mathnet.ru/eng/smj/v59/i1/p29
  • This publication is cited in the following 10 articles:
    1. Serikzhan A. Badaev, Nikolay A. Bazhenov, Birzhan S. Kalmurzayev, Manat Mustafa, “On diagonal functions for equivalence relations”, Arch. Math. Logic, 63:3-4 (2024), 259  crossref
    2. B. S. Kalmurzaev, N. A. Bazhenov, D. B. Alish, “On universal positive graphs”, Siberian Math. J., 64:1 (2023), 83–93  mathnet  crossref  crossref  mathscinet
    3. N. Bazhenov, B. Kalmurzayev, M. Zubkov, “A note on joins and meets for positive linear preorders”, Sib. elektron. matem. izv., 20:1 (2023), 1–16  mathnet  crossref
    4. B. S. Kalmurzaev, N. A. Bazhenov, M. A. Torebekova, “Ob indeksnykh mnozhestvakh dlya klassov pozitivnykh predporyadkov”, Algebra i logika, 61:1 (2022), 42–76  mathnet  crossref
    5. B. S. Kalmurzayev, N. A. Bazhenov, M. A. Torebekova, “Index Sets for Classes of Positive Preorders”, Algebra Logic, 61:1 (2022), 30  crossref
    6. S. A. Badaev, B. S. Kalmurzayev, N. K. Mukash, A. A. Khamitova, “Special classes of positive preorders”, Sib. elektron. matem. izv., 18:2 (2021), 1657–1666  mathnet  crossref
    7. S. A. Badaev, N. A. Bazhenov, B. S. Kalmurzaev, “The structure of computably enumerable preorder relations”, Algebra and Logic, 59:3 (2020), 201–215  mathnet  crossref  crossref  isi
    8. N. A. Bazhenov, B. S. Kalmurzaev, “Rogers semilattices for families of equivalence relations in the Ershov hierarchy”, Siberian Math. J., 60:2 (2019), 223–234  mathnet  crossref  crossref  isi  elib
    9. N. A. Bazhenov, B. S. Kalmurzaev, “Weakly precomplete equivalence relations in the Ershov hierarchy”, Algebra and Logic, 58:3 (2019), 199–213  mathnet  crossref  crossref  isi
    10. Badaev S.A., Kalmurzayev B.S., Kabylzhanova D.K., Abeshev K.Sh., “Universal Positive Preorders”, News Natl. Acad. Sci. Rep. Kazakhstan-Ser. Phys.-Math., 6:322 (2018), 49–53  crossref  mathscinet  isi
    Citing articles in Google Scholar: Russian citations, English citations
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