Siberian Mathematical Journal

, Volume 59, Issue 1, pp 22–30 | Cite as

On Dark Computably Enumerable Equivalence Relations

  • N. A. BazhenovEmail author
  • B. S. Kalmurzaev


We study computably enumerable (c.e.) relations on the set of naturals. A binary relation R on ω is computably reducible to a relation S (which is denoted by R c S) 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 ≤ 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 c F. As a consequence of this result, we construct an infinite increasing ≤ 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 ≤ c .


equivalence relation computably enumerable equivalence relation computable reducibility weakly precomplete equivalence relation computably enumerable order lo-reducibility 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Mal’tsev A. I., “Sets with complete numberings,” Algebra and Logic, vol. 2, no. 2, 353–378 (1963).Google Scholar
  2. 2.
    Ershov Yu. L., The Theory of Enumerations [Russian], Nauka, Moscow (1977).Google Scholar
  3. 3.
    Bernardi C. and Sorbi A., “Classifying positive equivalence relations,” J. Symb. Log., vol. 48, no. 3, 529–538 (1983).MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Lachlan A. H., “A note on positive equivalence relations,” Z. Math. Logik Grundlagen Math., Bd 33, 43–46 (1987).MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Badaev S. A., “On weakly pre-complete positive equivalences,” Sib. Math. J., vol. 32, no. 2, 321–323 (1991).MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Badaev S. and Sorbi A., “Weakly precomplete computably enumerable equivalence relations,” Math. Log. Q., vol. 62, no. 1–2, 111–127 (2016).MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Ershov Yu. L., “Positive equivalences,” Algebra and Logic, vol. 10, no. 6, 620–650 (1971).MathSciNetCrossRefGoogle Scholar
  8. 8.
    Gao S. and Gerdes P., “Computably enumerable equivalence relations,” Studia Log., vol. 67, no. 1, 27–59 (2001).MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Andrews U., Lempp S., Miller J. S., Ng K. M., San Mauro L., and Sorbi A., “Universal computably enumerable equivalence relations,” J. Symb. Log., vol. 79, no. 1, 60–88 (2014).MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Andrews U. and Sorbi A., “The complexity of index sets of classes of computably enumerable equivalence relations,” J. Symb. Log., vol. 81, no. 4, 1375–1395 (2016).MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Andrews U., Badaev S., and Sorbi A., “A survey on universal computably enumerable equivalence relations,” in: Computability and Complexity, Springer-Verlag, Cham, 2016, 418–451 (Lect. Notes Comput. Sci.; vol. 10010).Google Scholar
  12. 12.
    Gavryushkin A., Khoussainov B., and Stephan F., “Reducibilities among equivalence relations induced by recursively enumerable structures,” Theor. Comput. Sci., vol. 612, 137–152 (2016).MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Fokina E., Khoussainov B., Semukhin P., and Turetsky D., “Linear orders realized by c.e. equivalence relations,” J. Symb. Log., vol. 81, no. 2, 463–482 (2016).MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Ershov Yu. L. and Goncharov S. S., Constructive Models, ser. Siberian School of Algebra and Logic, Kluwer Academic/ Plenum Publishers, New York, etc. (2000).Google Scholar
  15. 15.
    Harizanov V. S., “Turing degrees of certain isomorphic images of computable relations,” Ann. Pure Appl. Logic, vol. 93, no. 1–3, 103–113 (1998).MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  1. 1.Sobolev Institute of MathematicsNovosibirsk State UniversityNovosibirskRussia
  2. 2.Al-Farabi Kazakh National UniversityAlmatyKazakhstan

Personalised recommendations