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Weakly Precomplete Equivalence Relations in the Ershov Hierarchy

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Algebra and Logic Aims and scope

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 E ≤cF. 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.

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References

  1. Yu. L. Eršov, “Theorie der Numerierungen. I,” Z. Math. Logik Grundl. Math., 19, No. 4, 289-388 (1973).

    Article  MathSciNet  Google Scholar 

  2. Yu. L. Eršov, “Theorie der Numerierungen. II,” Z. Math. Logik Grundl. Math., 21, No. 6, 473-584 (1975).

    Article  MathSciNet  Google Scholar 

  3. Yu. L. Ershov, “Positive equivalences,” Algebra and Logic, 10, No. 6, 378-394 (1971).

    Article  MathSciNet  Google Scholar 

  4. A. I. Mal’tsev, “Sets with complete numberings,” Algebra Logika, 2, No. 2, 4-29 (1963).

    MathSciNet  Google Scholar 

  5. Yu. L. Ershov, Numeration Theory [in Russian], Nauka, Moscow (1977).

    Google Scholar 

  6. C. Bernardi and A. Sorbi, “Classifying positive equivalence relations,” J. Symb. Log., 48, No. 3, 529-538 (1983).

    Article  MathSciNet  Google Scholar 

  7. A. H. Lachlan, “A note on positive equivalence relations,” Z. Math. Logik Grundlagen Math., 33, 43-46 (1987).

    Article  MathSciNet  Google Scholar 

  8. U. Andrews, S. Badaev, and A. Sorbi, “A survey on universal computably enumerable equivalence relations,” in Lect. Notes Comput. Sci., 10010, Springer, Cham (2017), pp. 418-451.

  9. S. A. Badaev, “On weakly pre-complete positive equivalences,” Sib. Math. J., x, No. 2, 321-323 (1991).

    Article  MathSciNet  Google Scholar 

  10. S. Badaev and A. Sorbi, “Weakly precomplete computably enumerable equivalence relations,” Math. Log. Q., 62, Nos. 1/2, 111-127 (2016).

    Article  MathSciNet  Google Scholar 

  11. N. A. Bazhenov and B. S. Kalmurzaev, “On dark computably enumerable equivalence relations,” Sib. Math. J., 59, No. 1, 22-30 (2018).

    Article  MathSciNet  Google Scholar 

  12. K. M. Ng and H. Yu, “On the degree structure of equivalence relations under computable reducibility,” submitted to Notre Dame J. Form. Log., 2018.

  13. U. Andrews and A. Sorbi, “Joins and meets in the structure of Ceers,” 2018; https://arxiv.org/abs/1802.09249.

  14. U. Andrews and A. Sorbi, “Jumps of computably enumerable equivalence relations,” Ann. Pure Appl. Log., 169, No. 3, 243-259 (2018).

    Article  MathSciNet  Google Scholar 

  15. S. Gao and P. Gerdes, “Computably enumerable equivalence relations,” Stud. Log., 67, No. 1, 27-59 (2001).

    Article  MathSciNet  Google Scholar 

  16. Yu. L. Ershov, “On a hierarchy of sets I,” Algebra and Logic, 7, No. 1, 25-43 (1968).

    Article  MathSciNet  Google Scholar 

  17. Yu. L. Ershov, “On a hierarchy of sets II,” Algebra and Logic, 7, No. 4, 212-232 (1968).

    Article  MathSciNet  Google Scholar 

  18. Yu. L. Ershov, “On a hierarchy of sets III,” Algebra and Logic, 9, No. 1, 20-31 (1970).

    Article  MathSciNet  Google Scholar 

  19. C. J. Ash and J. F. Knight, Computable Structures and the Hyperarithmetical Hierarchy, Stud. Logic Found. Math., 144, Elsevier, Amsterdam (2000).

  20. H. Rogers, Theory of Recursive Functions and Effective Computability, McGraw-Hill, New York (1967).

    MATH  Google Scholar 

  21. S. S. Goncharov and A. Sorbi, “Generalized computable numerations and nontrivial Rogers semilattices,” Algebra and Logic, 36, No. 6, 359-369 (1997).

    Article  MathSciNet  Google Scholar 

  22. S. A. Badaev and S. S. Goncharov, “Computability and numberings,” in New Computational Paradigms, S. B. Cooper, B. Lowe, and A. Sorbi (eds.), Springer, New York (2008), pp. 19-34.

  23. S. A. Badaev and S. S. Goncharov, “Theory of numberings: Open problems,” in Computability Theory and Its Applications, Cont. Math., 257, Am. Math. Soc., Providence, R.I. (2000), pp. 23-38.

  24. Yu. L. Ershov, “Theory of numberings,” in Handbook of Computability Theory, Stud. Log. Found. Math., 140, North-Holland, Amsterdam (1999), pp. 473-503.

  25. S. Ospichev, “Computable family of \( {\varSigma}_a^{-1} \) ss-sets without Friedberg numberings,” in 6th Conf. Comput. Europe, CiE 2010, Ponta Delgada, Azores (2010), pp. 311-315.

  26. E. Ianovski, R. Miller, K. M. Ng, and A. Nies, “Complexity of equivalence relations and preorders from computability theory,” J. Symb. Log., 79, No. 3, 859-881 (2014).

    Article  MathSciNet  Google Scholar 

  27. V. L. Selivanov, “On the Ershov hierarchy,” Sib. Math. J., 26, No. 1, 105-116 (1985).

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Sobolev Institute of Mathematics, pr. Akad. Koptyuga 4, Novosibirsk, 630090, Russia

    N. A. Bazhenov

  2. Novosibirsk State University, ul. Pirogova 1, Novosibirsk, 630090, Russia

    N. A. Bazhenov

  3. Al-Farabi Kazakh National University, Al-Farabi Ave. 71, Alma-Ata, Kazakhstan, 050038

    B. S. Kalmurzaev

Authors
  1. N. A. Bazhenov
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  2. B. S. Kalmurzaev
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Correspondence to N. A. Bazhenov.

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*Supported by KN MON RK, project No. AP 05131579.

**Supported by RFBR, project no. 17-301-50022 mol_nr.

Translated from Algebra i Logika, Vol. 58, No. 3, pp. 297-319, May-June, 2019.

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Bazhenov, N.A., Kalmurzaev, B.S. Weakly Precomplete Equivalence Relations in the Ershov Hierarchy. Algebra Logic 58, 199–213 (2019). https://doi.org/10.1007/s10469-019-09538-y

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