APPROXIMATIONS OF THE THEORIES OF STRUCTURES WITH ONE EQUIVALENCE RELATION

Recently, various methods similar to the “transfer principle” have been rapidly developing, where one property of a structure or pieces of this structure is satisfied in all infinite structures or in another algebraic structure. Such methods include smoothly approximable structures, holographic structures, almost sure theories, and pseudofinite structures approximable by finite structures. Pseudofinite structures are mathematical structures that resemble finite structures but are not actually finite. They are important in various areas of mathematics, including model theory and algebraic geometry. Pseudofinite structures are a fascinating area of mathematical logic that bridge the gap between finite and infinite structures. They allow studying infinite structures in ways that resemble finite structures, and they provide a connection to various other concepts in model theory. Further studying pseudofinite structures will continue to reveal new insights and applications in mathematics and beyond. Pseudofinite theory is a branch of mathematical logic that studies structures that are similar in some ways to finite structures, but can be infinitely large in other ways. It is an area of research that lies at the intersection of model theory and number theory and deals with infinite structures that share some properties with finite structures, such as having only finitely many elements up to isomorphism. A. Lachlan introduced the concept of smoothly approximable structures in order to change the direction of analysis from finite to infinite, that is, to classify large finite structures that seem to be smooth approximations to an infinite limit. The theory of pseudofinite structures is particularly relevant for studying equivalence relations. In this paper, we study the model-theoretic property of the theory of equivalence relations, in particular, the property of smooth approximability. Let L = {E}, where Е is an equivalence relation. We prove that an any ω-categorical L-structure M is smoothly approximable. We also prove that any infinite L-structure M is pseudofinite.

1. Markhabatov N.D. (2019) Ranks for Families of Permutation Theories, The Bulletin of Irkutsk State University. Series Mathematics, vol. 28, pp.85–94. https://doi.org/10.26516/1997-7670.2019.28.85

2. Markhabatov N.D. (2019) Pseudofiniteness of locally free algebras, Algebra and Model Theory 12, Collection of papers, eds. A.G. Pinus, E.N. Poroshenko, S.V. Sudoplatov. Novosibirsk: NSTU, pp. 41–46.

3. Kulpeshov B.Sh., Sudoplatov S.V. (2019) Ranks and approximations for families of ordered theories, Algebra and Model Theory 12, Collection of papers, eds. A.G. Pinus, E.N. Poroshenko, S.V. Sudoplatov. Novosibirsk: NSTU, pp. 32–40.

4. Pavlyuk I.I. (2020) Approximations for Theories of Abelian Groups, eds. I.I. Pavlyuk, S.V. Sudoplatov, Mathematics and Statistics, vol. 8, no. 2, pp. 220–224. https://doi.org/10.13189/ms.2020.080218

5. Pavlyuk I.I., Sudoplatov S.V. (2019) Ranks for families of theories of abelian groups, The Bulletin of Irkutsk State University, Series Mathematics, vol., 28, pp. 95–112. https://doi.org/10.26516/1997-7670.2019.28.95

6. Markhabatov N.D., Sudoplatov S.V. (2022) Approximations Of Regular Graphs, Herald of the Kazakh-British Technical University, 19:1, pp. 44–49 https://doi.org/10.55452/1998-6688-2022-19-1-44-49

7. Tussupov D.A. (2016) Isomorphisms and Algorithmic Properties of Structures with Two Equivalences, Algebra Logic 55, pp. 50–57. https://doi.org/10.1007/s10469-016-9375-8

8. Ax J. (1968) The Elementary Theory of Finite Fields, Annals of Mathematics, vol. 88, no. 2, Annals of Mathematics, pp. 239–71. https://doi.org/10.2307/1970573

9. Kantor W.M., Liebeck M.W. and Macpherson H.D. (1989) N0-categorical structures smoothly approximated by finite substructures, Proceedings of the London Mathematical Society, vol. 59, pp. 439–463. https://doi.org/10.1112/plms/s3-59.3.439

10. Macpherson D. (1997). Homogeneous and Smoothly Approximated Structures. In: Hart, B.T., Lachlan, A.H., Valeriote, M.A. (eds) Algebraic Model Theory. NATO ASI Series, vol 496. Springer, Dordrecht. https://doi.org/10.1007/978-94-015-8923-9_7 ,

11. Cherlin G., and E. Hrushovski (2003) Finite Structures with Few Types, vol. 152, Annals of Mathematics Studies, Princeton University Press, Princeton.

12. Hrushovski E. (1993) Finite Structures with Few Types. In: Sauer N.W., Woodrow R.E., Sands B. (eds) Finite and Infinite Combinatorics in Sets and Logic. NATO ASI Series (Series C: Mathematical and Physical Sciences), vol 411. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-2080-7

13. Markhabatov N.D., Sudoplatov S.V. (2021) Ranks for families of all theories of given languages, Eurasian Math. J., 12:2, pp. 52–58. https://doi.org/10.32523/2077-9879-2021-12-2-52-58

14. Sudoplatov S.V. (2021) Ranks for families of theories and their spectra~/ S.~V.~Sudoplatov// Lobachevskii J. Math 42, pp. 2959–2968. https://doi.org/10.1134/S1995080221120313

15. Gaifman H. (1964) Concerning measures in first-order calculi, Israel J. Math. 2, pp. 1–18. https://doi.org/10.1007/BF02759729

16. Lynch J. (1980) Almost sure theories, Annals of Mathematical Logic, 18, pp. 91–135. https://doi.org/10.1016/0003-4843(80)90014-5

17. Ax J., Kochen S. (1965) Diophantine problems over local fields I. Amer. J. Math. 87, pp. 605–630. https://doi.org/10.2307/2373065

18. Sudoplatov S.V. (2020) Approximations of theories, Siberian Electronic Mathematical Reports, vol. 17, pp. 715–725. https://doi.org/10.33048/semi.2020.17.049