Siberian Mathematical Journal

, Volume 58, Issue 3, pp 553–558 | Cite as

Minimal generalized computable enumerations and high degrees

  • M. Kh. FaizrahmanovEmail author


We establish that the set of minimal generalized computable enumerations of every infinite family computable with respect to a high oracle is effectively infinite. We find some sufficient condition for enumerations of the infinite families computable with respect to high oracles under which there exist minimal generalized computable enumerations that are irreducible to the enumerations.


generalized computable enumeration minimal enumeration high set Low2 set arithmetic enumeration 


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  1. 1.
    Goncharov S. S. and Sorbi A., “Generalized computable numerations and nontrivial Rogers semilattices,” Algebra and Logic, vol. 36, no. 6, 359–369 (1997).MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Badaev S. A. and Goncharov S. S., “Rogers semilattices of families of arithmetic sets,” Algebra and Logic, vol. 40, no. 5, 283–291 (2001).MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Badaev S. A. and Podzorov S. Yu., “Minimal coverings in the Rogers semilattices of Σn 0-computable numberings,” Sib. Math. J., vol. 43, no. 4, 616–622 (2002).CrossRefzbMATHGoogle Scholar
  4. 4.
    Badaev S. A., Goncharov S. S., and Sorbi A., “Elementary theories for Rogers semilattices,” Algebra and Logic, vol. 44, no. 3, 143–147 (2005).MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Badaev S. A., Goncharov S. S., and Sorbi A., “Isomorphism types of Rogers semilattices for families from different levels of the arithmetical hierarchy,” Algebra and Logic, vol. 45, no. 6, 361–370 (2006).MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Ershov Yu. L., Theory of Numberings [Russian], Nauka, Moscow (1977).zbMATHGoogle Scholar
  7. 7.
    Ershov Yu. L., “Theory of numberings,” in: Handbook of Computability Theory (E. R. Griffor, ed.), Elsevier, Amsterdam, 1999, 473–503 (Stud. Logic Found. Math.; vol. 140).CrossRefGoogle Scholar
  8. 8.
    Soare R. I., Recursively Enumerable Sets and Degrees: A Study of Computable Functions and Computably Generated Sets, Springer-Verlag, Berlin, Heidelberg, New York, London, Paris, and Tokyo (1987).CrossRefzbMATHGoogle Scholar
  9. 9.
    Faizrahmanov M. Kh., “Universal generalized computable numerations and hyperimmunity,” Algebra and Logic (to be published).Google Scholar

Copyright information

© Pleiades Publishing, Ltd. 2017

Authors and Affiliations

  1. 1.Kazan (Volga Region) Federal UniversityKazanRussia

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