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Computability and Universal Determinability of Negatively Representable Models

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Abstract

It has been established that a negative representable model is computable if and only if its standard enrichment with constants is isomorphically embedded in any model of a suitable computable enumerated set of universal sentences implemented in this model. It is shown that for computable enumerable sets of existential sentences this statement is incorrect.

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REFERENCES

  1. S. S. Goncharov and Yu. L. Ershov, Constructive Models (Nauchnaya Kniga, Novosibirsk, 1999; Springer, New York, 2000).

  2. Yu. L. Ershov, Decidability Problems and Constructive Models (Nauka, Moscow, 1980)

    Google Scholar 

  3. S. S. Goncharov, “Models of data and languages for their descriptions,” Vychisl. Sist. 107, 52–70 (1985).

    MathSciNet  MATH  Google Scholar 

  4. N. Kh. Kasymov, “Recursively separable enumerated algebras,” Russ. Math. Surv. 51 (3), 509–538 (1996). https://doi.org/10.1070/RM1996v051n03ABEH002913

    Article  MathSciNet  MATH  Google Scholar 

  5. A. I. Mal’tsev, “Constructive algebras. I,” Russ. Math. Surv. 16 (3), 77–129 (1961). https://doi.org/10.1070/RM1961v016n03ABEH001120

    Article  MathSciNet  MATH  Google Scholar 

  6. A. I. Mal’tsev, Algebraic Systems (Nauka, Moscow, 1970) [in Russian].

    MATH  Google Scholar 

  7. I. R. Soar, Computably Enumerable Sets and Degrees (Kazan Mathematical Society, Kazan, 2000) [in Russian].

    Google Scholar 

  8. N. Kh. Kasymov and A. S. Morozov, “Logical aspects of the theory of abstract data types,” Vychisl. Sist. 122, 73–96 (1987).

    MATH  Google Scholar 

  9. J. A. Bergstra and J.V. Tucker, “A characterization of computable data types by means of a finite equational specification method,” in Automata, Languages and Programming, ICALP 1980, Ed. by J. Bakker and J. van Leeuwen, Lecture Notes in Computer Science, Vol. 85 (Springer, Berlin, 1980), pp. 76–90. https://doi.org/10.1007/3-540-10003-2_61

  10. N. Kh. Kasymov, “Algebras with residually finite positively presented expansions,” Algebra Logika 26 (6), 715–730 (1987).

    Article  MathSciNet  MATH  Google Scholar 

  11. N. Kh. Kasymov, “Separation axioms and partitions of the set of natural numbers,” Sib. Math. J. 34 (3) 468–471 (1993). https://doi.org/10.1007/BF00971221

    Article  MATH  Google Scholar 

  12. N. Kh. Kasymov, “Enumerated algebras with uniformly recursive-separable classes,” Sib. Math. J. 34 (5), 869–882 (1993). https://doi.org/10.1007/BF00971403

    Article  MATH  Google Scholar 

  13. N. Kh. Kasymov, “Homomorphisms onto effectively separable algebras,” Sib. Math. J. 57 (1), 36–50 (2016). https://doi.org/10.1134/S0037446616010055

    Article  MathSciNet  MATH  Google Scholar 

  14. N. Kh. Kasymov and F. N. Ibragimov, “Separable enumerations of division rings and effective embeddability of rings therein,” Sib. Math. J. 60 (1), 62–70 (2019). https://doi.org/10.1134/S0037446619010075

    Article  MathSciNet  MATH  Google Scholar 

  15. N. Kh. Kasymov, “Homomorphisms on negative algebras,” Algebra Logic 31 (2), 81–89 (1992). https://doi.org/10.1007/BF02259847

    Article  MathSciNet  MATH  Google Scholar 

  16. N. Kh. Kasymov and F. N. Ibragimov, “Computably separable models,” J. Math. Sci. 264 (6), 746–767 (2022). https://doi.org/10.1007/s10958-022-06033-1

    Article  MathSciNet  MATH  Google Scholar 

  17. N. Kh. Kasymov and I. A. Khodzhamuratova, “Topological spaces over algorithmic representations of universal algebras,” J. Math. Sci. 245 (3), 311–322 (2020). https://doi.org/10.1007/s10958-020-04692-6

    Article  MATH  Google Scholar 

  18. N. Kh. Kasymov, “Algebras over negative equivalences,” Algebra Logic 33 (1), 76–80 (1994). https://doi.org/10.1007/BF00739416

    Article  MathSciNet  MATH  Google Scholar 

  19. N. Kh. Kasymov, “Positive algebras with congruences of finite index,” Algebra Logic 30 (3), 190–199 (1991). https://doi.org/10.1007/BF01978852

    Article  MathSciNet  MATH  Google Scholar 

  20. N. Kh. Kasymov, “Positive algebras with countable congruence lattices,” Algebra Logic 31 (1), 12–23 (1992). 1992. https://doi.org/10.1007/BF02259854

  21. Yu. L. Ershov, Theory of Numberings (Nauka, Moscow, 1977) [in Russian].

    MATH  Google Scholar 

  22. N. Kh. Kasymov, R. N. Dadazhanov, and S. K. Djavliev, “Structures of degrees of negative representations of linear orders,” Russ. Math. 65 (12), 27–46 (2021). https://doi.org/10.3103/S1066369X21120045

    Article  MATH  Google Scholar 

  23. B. Khoussainov, T. Slaman, and P. Semukhin, “\(\Pi _{1}^{0}\)-presentasions of algebras,” Arch. Math. Logic 45 (6), 769–781 (2006). https://doi.org/10.1007/s00153-006-0013-3

    Article  MathSciNet  Google Scholar 

  24. N. Kh. Kasymov and R. N. Dadazhanov, “Negative dense linear orders,” Sib. Math. J. 58 (6), 1015–1033 (2017). https://doi.org/10.1134/S0037446617060118

    Article  MathSciNet  MATH  Google Scholar 

  25. N. Kh. Kasymov and A. S. Morozov, “Definability of linear orders over negative equivalences,” Algebra Logic 55 (1), 24–37 (2016). https://doi.org/10.1007/s10469-016-9373-x

    Article  MathSciNet  MATH  Google Scholar 

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  1. National University of Uzbekistan, 4 University str., 100174, Tashkent, Republic of Uzbekistan

    R. N. Dadazhanov

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  1. R. N. Dadazhanov
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Dadazhanov, R.N. Computability and Universal Determinability of Negatively Representable Models. Russ Math. 66, 16–24 (2022). https://doi.org/10.3103/S1066369X22100036

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