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 Sibirsk. Mat. Zh., 2018, Volume 59, Number 1, Pages 29–40 (Mi smj2951)

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

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 ìîë_à_äê Ministry of Education and Science of the Republic of Kazakhstan ÃÔ4/3952 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).

DOI: https://doi.org/10.17377/smzh.2018.59.103

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English version:
Siberian Mathematical Journal, 2018, 59:1, 22–30

Bibliographic databases:

Document Type: Article
UDC: 510.57
MSC: 35R30

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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