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Algebra and Logic

, Volume 58, Issue 3, pp 199–213 | Cite as

Weakly Precomplete Equivalence Relations in the Ershov Hierarchy

  • N. A. BazhenovEmail author
  • B. S. Kalmurzaev
Article
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We study the computable reducibility ≤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 \( {\varPi}_a^{-1} \) -universal equivalence relation and a weakly precomplete \( {\varSigma}_a^{-1} \) - universal equivalence relation. We prove that for any \( {\varSigma}_a^{-1} \) equivalence relation E, there is a weakly precomplete \( {\varSigma}_a^{-1} \) equivalence relation F such that EcF. For finite levels \( {\varSigma}_m^{-1} \) in the Ershov hierarchy at which m = 4k +1 or m = 4k +2, it is shown that there exist infinitely many ≤c-degrees containing weakly precomplete, proper \( {\varSigma}_m^{-1} \) equivalence relations.

Keywords

Ershov hierarchy equivalence relation computable reducibility universal equivalence relation weakly precomplete equivalence relation 

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© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Sobolev Institute of MathematicsNovosibirskRussia
  2. 2.Novosibirsk State UniversityNovosibirskRussia
  3. 3.Al-Farabi Kazakh National UniversityAlma-AtaKazakhstan

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