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Algebra Logika, 2019, Volume 58, Number 3, Pages 297–319 (Mi al896)  

This article is cited in 2 scientific papers (total in 2 papers)

Weakly precomplete equivalence relations in the Ershov hierarchy

N. A. Bazhenovab, B. S. Kalmurzaevc

a Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, Novosibirsk
b Novosibirsk State University
c Al-Farabi Kazakh National University

Abstract: We study the computable reducibility $\leq_c$ for equivalence relations in the Ershov hierarchy. For an arbitrary notation $a$ for a nonzero computable ordinal, it is stated that there exist a $\Pi^{-1}_a$-universal equivalence relation and a weakly precomplete $\Sigma^{-1}_a$-universal equivalence relation. We prove that for any $\Sigma^{-1}_a$ equivalence relation $E$, there is a weakly precomplete $\Sigma^{-1}_a$ equivalence relation $F$ such that $E\leq_c F$. For finite levels $\Sigma^{-1}_m$ in the Ershov hierarchy at which $m=4k+1$ or $m=4k+2$, it is shown that there exist infinitely many $\leq_c$-degrees containing weakly precomplete, proper $\Sigma^{-1}_m$ equivalence relations.

Keywords: Ershov hierarchy, equivalence relation, computable reducibility, universal equivalence relation, weakly precomplete equivalence relation.

Funding Agency Grant Number
Ministry of Education and Science of the Republic of Kazakhstan AP 05131579
Russian Foundation for Basic Research 17-301-50022_мол_нр
*Supported by KN MON RK, project No. AP 05131579. **Supported by RFBR, project no. 17-301-50022 mol_nr.


DOI: https://doi.org/10.33048/alglog.2019.58.301

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English version:
Algebra and Logic, 2019, 58:3, 199–213

Bibliographic databases:

UDC: 510.54
Received: 11.04.2018
Revised: 24.09.2019

Citation: N. A. Bazhenov, B. S. Kalmurzaev, “Weakly precomplete equivalence relations in the Ershov hierarchy”, Algebra Logika, 58:3 (2019), 297–319; Algebra and Logic, 58:3 (2019), 199–213

Citation in format AMSBIB
\Bibitem{BazKal19}
\by N.~A.~Bazhenov, B.~S.~Kalmurzaev
\paper Weakly precomplete equivalence relations in the Ershov hierarchy
\jour Algebra Logika
\yr 2019
\vol 58
\issue 3
\pages 297--319
\mathnet{http://mi.mathnet.ru/al896}
\crossref{https://doi.org/10.33048/alglog.2019.58.301}
\transl
\jour Algebra and Logic
\yr 2019
\vol 58
\issue 3
\pages 199--213
\crossref{https://doi.org/10.1007/s10469-019-09538-y}
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\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85074823099}


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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. I. Yu. Mogilnykh, “Perfect codes from $\mathrm{PGL}(2,5)$ in Star graphs”, Sib. elektron. matem. izv., 17 (2020), 534–539  mathnet  crossref
    2. N. A. Bazhenov, M. Mustafa, L. San Mauro, M. M. Yamaleev, “Minimal equivalence relations in hyperarithmetical and analytical hierarchies”, Lobachevskii J. Math., 41:2, SI (2020), 145–150  crossref  mathscinet  zmath  isi  scopus
  • Алгебра и логика Algebra and Logic
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