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The Rogers Semilattices of Generalized Computable Enumerations

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Abstract

We study the cardinality and structural properties of the Rogers semilattice of generalized computable enumerations with arbitrary noncomputable oracles and oracles of hyperimmune Turing degree. We show the infinity of the Rogers semilattice of generalized computable enumerations of an arbitrary nontrivial family with a noncomputable oracle. In the case of oracles of hyperimmune degree we prove that the Rogers semilattice of an arbitrary infinite family includes an ideal without minimal elements and establish that the top, if present, is a limit element under the condition that the family contains the inclusion-least set.

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References

  1. Goncharov S. S. and Sorbi A., “Generalized computable numerations and nontrivial Rogers semilattices,” Algebra and Logic, vol. 36, no. 6, 359–369 (1997).

    Article  MathSciNet  MATH  Google Scholar 

  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).

    Article  MathSciNet  MATH  Google Scholar 

  3. Podzorov S. Yu., “Dual covers of the greatest element of the Rogers semilattice,” Siberian Adv. Math., vol. 15, no. 2, 104–114 (2004).

    MathSciNet  MATH  Google Scholar 

  4. Podzorov S. Yu., “Local structure of Rogers semilattices of Σn 0-computable numberings,” Algebra and Logic, vol. 44, no. 2, 82–94 (2005).

    Article  MathSciNet  MATH  Google Scholar 

  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).

    Article  MathSciNet  MATH  Google Scholar 

  6. Badaev S. A. and Goncharov S. S., “Generalized computable universal numberings,” Algebra and Logic, vol. 53, no. 5, 355–364 (2014).

    Article  MathSciNet  MATH  Google Scholar 

  7. Issakhov A. A., “Ideals without minimal elements in Rogers semilattices,” Algebra and Logic, vol. 54, no. 3, 197–203 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  8. Faizrahmanov M. Kh., “Universal generalized computable numerations and hyperimmunity,” Algebra and Logic (to be published).

  9. Faizrahmanov M. Kh., “Minimal generalized computable enumerations and high degrees,” Sib. Math. J., vol. 58, no. 3, 553–558 (2017).

    Article  MathSciNet  MATH  Google Scholar 

  10. Odifreddi P., Classical Recursion Theory. Vol. 1: The Theory of Functions and Sets of Natural Numbers, Elsevier, Amsterdam (1992).

    MATH  Google Scholar 

  11. Ershov Yu. L., The Theory of Enumerations [Russian], Nauka, Moscow (1977).

    Google Scholar 

  12. Ershov Yu. L., “Theory of numberings,” in: E. R. Griffor, ed. Handbook of Computability Theory, Elsevier, Amsterdam, 1999, 473–503 (Stud. Logic Found. Math.; 140.

    Chapter  Google Scholar 

  13. Ershov Yu. L., “Enumeration of families of general recursive functions,” Sib. Math. J., vol. 8, no. 5, 771–778 (1967).

    Article  MATH  Google Scholar 

  14. Khutoretskii A. B., “On the cardinality of the upper semilattice of computable enumerations,” Algebra and Logic, vol. 10, no. 5, 348–352 (1971).

    Article  MathSciNet  Google Scholar 

  15. Selivanov V. L., “Two theorems on computable numberings,” Algebra and Logic, vol. 15, no. 4, 297–306 (1976).

    Article  MATH  Google Scholar 

  16. Stephan S., “On the structures inside truth-table degrees,” J. Symb. Log., vol. 66, no. 2, 731–770 (2001).

    Article  MathSciNet  MATH  Google Scholar 

  17. Badaev S. A., “A problem of S. S. Goncharov,” Sib. Math. J., vol. 32, no. 3, 532–534 (1991).

    Article  MathSciNet  MATH  Google Scholar 

  18. Badaev S. A., “Minimal enumerations,” in: Mathematical Logic and Algorithm Theory. Tr. Inst. Mat. [Russian], 1993, vol. 25, 3–34.

    MathSciNet  MATH  Google Scholar 

  19. Khutoretskii A. B., “Two existence theorems for computable numerations,” Algebra and Logic, vol. 8, no. 4, 277–282 (1969).

    Article  MathSciNet  Google Scholar 

  20. V’yugin V. V., “On some examples of upper semilattices of computable enumerations,” Algebra and Logic, vol. 12, no. 3, 287–296 (1973).

    Article  MATH  Google Scholar 

  21. Martin D. and Miller W., “The degrees of hyperimmune sets,” Z. Math. Logik Grundlag. Math., Bd 14, 159–166 (1968).

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Kazan (Volga Region) Federal University, Kazan, Russia

    M. Kh. Faizrahmanov

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  1. M. Kh. Faizrahmanov
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Correspondence to M. Kh. Faizrahmanov.

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Original Russian Text Copyright © 2017 Faizrahmanov M.Kh.

The author was supported by the subsidy of the government task for Kazan (Volga Region) Federal University (Grant 1.1515.2017/4.6) and the Russian Foundation for Basic Research (Grant 15–01–08252).

Kazan. Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 58, No. 6, pp. 1418–1427, November–December, 2017

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Faizrahmanov, M.K. The Rogers Semilattices of Generalized Computable Enumerations. Sib Math J 58, 1104–1110 (2017). https://doi.org/10.1134/S0037446617060192

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  • DOI: https://doi.org/10.1134/S0037446617060192

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