Bulletin of National University of Uzbekistan: Mathematics and Natural Sciences


It is established that any effectively separable multi-sorted positively representable model with an effectively separable representation kernel has an enrichment that is the only (up to isomorphism) model constructed from constants for a suitable computably enumerable set of sentences.

First Page


Last Page



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

2. Goncharov S. S., Ershov Yu. L. Constructive Models (ser. Siberian School of Algebra and Logic). Kluwer Academic/Plenum Publishers, New York, etc. (2000).

3. Soare R. I. Recursively Enumerable Sets and Degrees: A Study of Computable Functions and Computably Generated Sets. Springer-Verlag, Berlin, Heidelberg (1987).

4. Bergstra J. A., Tucker J. V. A characterization of computable data types by means of a finite, equational specification method. Lecture Notes in Comput. Sci., Vol. 85, 76–90 (1980).

5. Goncharov S. S. Models of data and languages for their descriptions. Vychisl. Sistemy, Vol. 107, 52–77 (1985) [in Russian].

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

7. Kasymov N. Kh. Recursively separable enumerated algebras. Russian Math. Surveys, Vol. 51, Issue 3, 509–538 (1996).

8. Kasymov N. Kh., Dadazhanov R. N., Ibragimov F. N. Separable algorithmic representations of classical systems and their applications. Science – Technology – Education – Mathematics – Medicine, CMFD, Vol. 67, No. 4, PFUR, M., 707–754 (2021).

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

10. Kasymov N. Kh. Algebras with residually finite positively presented enrichments. Algebra and Logic, Vol. 26, No. 6, 715–730 (1987).

11. Khoussainov B. M. Randomness, computability, and algebraic specifications. Annals of Pure and Applied Logic, Vol. 91, Issue 1, 1–15 (1998).

12. Khoussainov B. M., Miasnikov A. G. Finitely presented expansions of groups, semigroups, and algebras. Trans. Amer. Math. Soc., Vol. 366, Issue 3, 1455–1474 (2014).

13. Kasymov N. Kh. Separation axioms and partitions of the set of natural numbers. Sib. Mat. Zh., Vol. 34, No. 3, 81–85 (1993).

14. Kasymov N. Kh. Enumerated algebras with uniformly recursive-separable classes. Sib. Mat. Zh., Vol. 34, No. 5, 85–102 (1993).

15. Kasymov N. Kh. Homomorphisms onto negative algebras. Algebra and Logic, Vol. 31, No. 2, 132–144 (1992).

16. Kasymov N. Kh. Algebras over negative equivalences. Algebra and Logic, Vol. 33, No. 1, 46–48 (1994).

17. Kasymov N. Kh., Positive algebras with congruences of finite index, Algebra and Logic, Vol. 30, No. 3, 293–305 (1991).

18. Kasymov N. Kh., Positive algebras with countable congruence lattices, Algebra and Logic, Vol. 31, No. 1, 21–37 (1992).

19. Khoussainov B. M., Slaman T., Semukhin P. Î 01-Presentasions of algebras. Archive for Mathematical Logic, Vol. 45, Issue 6, 769–781 (2006).

20. Kasymov N. K., Dadazhanov R. N., Djavliev S. K. Structures of degrees of negative representations of linear orders. Russ Math., Vol. 65, 27–46 (2021).

21. Kasymov N. Kh., Dadazhanov R. N. Negative dense linear orders. Sib. Math. J., Vol. 58, No. 6, 1015–1033 (2017).

22. Kasymov N. Kh., Morozov A. S. Definability of linear orders over negative equivalences. Algebra and Logic, Vol. 55, No. 1, 24–37 (2016).

23. Kasymov N. Kh. Homomorphisms onto effectively separable algebras. Sib. Math. J., Vol. 57, No. 1, 36–50 (2016).

24. Kasymov N. Kh., Morozov A. S., Khodzhamuratova I. A. T1-separable numberings of subdirectly indecomposable algebras. Algebra and Logic, Vol. 60, Issue 4, 263–278 (2021).

25. Andrews U., Sorbi A. Joins and meets in the structure of ceers. Computability, Vol. 8, Issue 3–4, 193–241 (2019).

26. Andrews U., Belin D., San Mauro L. On the structure of computable reducibility on equivalence relations of natural numbers. J. Symb. Logic (published online), doi: 10.1017/jsl.2022.28, 1–26.

27. Dadazhanov R. N. Computability and universal determinability of negatively representable models. Russ Math., Vol. 66, 16–24 (2022).



To view the content in your browser, please download Adobe Reader or, alternately,
you may Download the file to your hard drive.

NOTE: The latest versions of Adobe Reader do not support viewing PDF files within Firefox on Mac OS and if you are using a modern (Intel) Mac, there is no official plugin for viewing PDF files within the browser window.