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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 $\omega$ is computably reducible to a relation $S$ (which is denoted by $R\leq_cS$) 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 $\leq_c$ 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 $E\leq_cF$. As a consequence of this result, we construct an infinite increasing $\leq_c$-chain of weakly precomplete dark c.e. equivalence relations. We also show the existence of a universal c.e. linear order with respect to $\leq_c$.
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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\by N.~A.~Bazhenov, B.~S.~Kalmurzaev
\paper On dark computably enumerable equivalence relations
\jour Sibirsk. Mat. Zh.
\yr 2018
\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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\transl
\jour Siberian Math. J.
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\vol 59
\issue 1
\pages 22--30
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  • This publication is cited in the following 10 articles:
    Citing articles in Google Scholar: Russian citations, English citations
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    Сибирский математический журнал Siberian Mathematical Journal
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