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«THE BULLETIN OF IRKUTSK STATE UNIVERSITY». SERIES «MATHEMATICS»
«IZVESTIYA IRKUTSKOGO GOSUDARSTVENNOGO UNIVERSITETA». SERIYA «MATEMATIKA»
ISSN 1997-7670 (Print)
ISSN 2541-8785 (Online)

List of issues > Series «Mathematics». 2023. Vol 45

Ranks, Spectra and Their Dynamics for Families of Constant Expansions of Theories

Author(s)
Beibut Sh. Kulpeshov1,2,3, Sergey V. Sudoplatov2,4

1Kazakh British Technical University, Almaty, Kazakhstan

2Novosibirsk State Technical University, Novosibirsk, Russian Federation

3Institute of Mathematics and Mathematical Modeling, Almaty, Kazakhstan

4Sobolev Institute of Mathematics SB RAS, Novosibirsk, Russian Federation

Abstract

Constant or nonessential extensions of elementary theories provide a productive tool for the study and structural description of models of these theories, which is widely used in Model Theory and its applications, both for various stable and ordered theories, countable and uncountable theories, algebraic, geometric and relational structures and theories. Families of constants are used in Henkin’s classical construction of model building for consistent families of formulas, for the classification of uncountable and countable models of complete theories, and for some dynamic possibilities of countable spectra of ordered Ehrenfeucht theories.

The paper describes the possibilities of ranks and degrees for families of constant extensions of theories. Rank links are established for families of theories with CantorBendixson ranks for given theories. It is shown that the 𝑒-minimality of a family of constant expansions of the theory is equivalent to the existence and uniqueness of a nonprincipal type with a given number of variables. In particular, for strongly minimal theories this means that the non-principal 1-type is unique over an appropriate tuple. Relations between 𝑒-spectra of families of constant expansions of theories and ranks and degrees are established. A model-theoretic characterization of the existence of the least generating set is obtained.

It is also proved that any inessential finite expansion of an o-minimal Ehrenfeucht theory preserves the Ehrenfeucht property, and this is true for constant expansions of dense spherically ordered theories. For the expansions under consideration, the dynamics of the values of countable spectra is described.

About the Authors

Beibut Sh. Kulpeshov, Dr. Sci. (Phys.-Math.), Prof., Kazakh British Technical University, Almaty, 050000, Kazakhstan, b.kulpeshov@kbtu.kz; Novosibirsk State Technical University, Novosibirsk, 630073, Russian Federation, kulpeshov@corp.nstu.ru; Institute of Mathematics and Mathematical Modeling, Almaty, 050010, Kazakhstan, kulpesh@mail.ru

Sergey V. Sudoplatov, Dr. Sci. (Phys.–Math.), Prof., Sobolev Institute of Mathematics SB RAS, Novosibirsk, 630090, Russian Federation; Novosibirsk State Technical University, Novosibirsk, 630073, Russian Federation, sudoplat@math.nsc.ru

For citation
Kulpeshov B. Sh., Sudoplatov S. V. Ranks, Spectra and Their Dynamics for Families of Constant Expansions of Theories. The Bulletin of Irkutsk State University. Series Mathematics, 2023, vol. 45, pp. 121–137. https://doi.org/10.26516/1997-7670.2023.45.121
Keywords
family of theories, rank, degree, constant expansion, Ehrenfeucht theory, ordered theory, spherical theory
UDC
510.67
MSC
03C30, 03C15, 03C50
DOI
https://doi.org/10.26516/1997-7670.2023.45.121
References
  1. Baizhanov B.S. Expansion of a model of a weakly o-minimal theory by a family of unary predicates. The Journal of Symbolic Logic, 2001, vol. 66, no. 3, pp. 1382–1414. https://doi.org/10.2307/2695114
  2. Chang C.C., Keisler H.J. Model Theory. Mineola, New York, Dover Publications, Inc., 2012. 650 p.
  3. Ershov Yu.L., Palyutin E.A. Mathematical Logic. Moscow, Fizmatlit Publ., 2011. 356 p. (in Russian)
  4. Fra¨ıss´e R. Theory of relations. Amsterdam, North-Holland, 1986, 451 p.
  5. Handbook of Mathematical Logic, ed. J. Barwise. Moscow, Nauka Publ., 1982, vol. 1: Model Theory, 392 p. (in Russian)
  6. Hart B., Hrushovski E.,Laskowski M.S. The uncountable spectra of countable theories. Annals of Mathematics, 2000, vol. 152, no. 1, pp. 207–257. https://doi.org/10.2307/2661382
  7. Henkin L. The completeness of the first-order functional calculus. The Journal of Symbolic Logic, 1949, vol. 14, no. 3, pp. 159–166. https://doi.org/10.2307/2267044
  8. Hodges W. Model Theory. Cambridge, Cambridge University Press, 1993, 772 p.
  9. Kulpeshov B.Sh., Sudoplatov S.V. Vaught’s conjecture for quite o-minimal theories. Annals of Pure and Applied Logic, 2017, vol. 168, iss. 1, pp. 129–149. https://doi.org/10.1016/j.apal.2016.09.002
  10. Kulpeshov B.Sh., Sudoplatov S.V. Properties of ranks for families of strongly minimal theories. Siberian Electronic Mathematical Reports, 2022, vol. 19, no. 1, pp. 120–124. https://doi.org/10.33048/semi.2022.19.011
  11. Kulpeshov B.Sh., Sudoplatov S.V. Spherical orders, properties and countable spectra of their theories. Siberian Electronic Mathematical Reports, 2023, vol. 20, no. 2, pp. 588–599. https://doi.org/10.33048/semi.2023.20.034
  12. Marker D. Model Theory: An Introduction, New York, Springer-Verlag, 2002. — Graduate texts in Mathematics. Vol. 217. 342 p.
  13. Markhabatov N.D., Sudoplatov S.V. Ranks for families of all theories of given languages. Eurasian Mathematical Journal, 2021, vol. 12, no. 2, pp. 52–58. https://doi.org/10.32523/2077-9879-2021-12-2-52-58
  14. Markhabatov N.D., Sudoplatov S.V. Definable subfamilies of theories, related calculi and ranks. Siberian Electronic Mathematical Reports, 2020, vol. 17, pp. 700–714. https://doi.org/10.33048/semi.2020.17.048
  15. Markhabatov N.D. Ranks for families of permutation theories. The Bulletin of Irkutsk State University. Series Mathematics, 2019, vol. 28, pp. 86–95. https://doi.org/10.26516/1997-7670.2019.28.85
  16. Mayer L.L. Vaught’s conjecture for o-minimal theories. The Journal of Symbolic Logic, 1988, vol. 53, no. 1, pp. 146–159. https://doi.org/10.2307/2274434
  17. Omarov B. Nonessential extensions of complete theories. Algebra and Logic, 1983, vol. 22, no. 5, pp. 390–397. https://doi.org/10.1007/BF01982116
  18. Pavlyuk In.I., Sudoplatov S.V. Ranks for families of theories of abelian groups. The Bulletin of Irkutsk State University. Series Mathematics, 2019, vol. 28, pp. 96–113. https://doi.org/10.26516/1997-7670.2019.28.95
  19. Pavlyuk In.I., Sudoplatov S.V. Formulas and properties for families of theories of abelian groups. The Bulletin of Irkutsk State University. Series Mathematics, 2021, vol. 36, pp. 95–109. https://doi.org/10.26516/1997-7670.2021.36.95
  20. Pillay A. Geometric Stability Theory. Oxford, Clarendon Press, 1996, 361 p.
  21. Poizat B.P. Cours de th´eorie des mod`eles. Villeurbane, Nur Al-Mantiq WalMa’rifah, 1985, 444 p.
  22. Shelah S. Classification theory and the number of non-isomorphic models. Amsterdam, North-Holland, 1990, 705 p.
  23. Sudoplatov S.V., Tanovi´c P. Semi-isolation and the strict order property. Notre Dame Journal of Formal Logic, 2015, vol. 56, no. 4, pp. 555–572. https://doi.org/10.1215/00294527-3153579
  24. Sudoplatov S.V. Closures and generating sets related to combinations of structures. The Bulletin of Irkutsk State University. Series Mathematics, 2016, vol. 16, pp. 131–144.
  25. Sudoplatov S.V. Families of language uniform theories and their generating sets. The Bulletin of Irkutsk State University. Series Mathematics, 2016, vol. 17, pp. 62–76.
  26. Sudoplatov S.V. Combinations related to classes of finite and countably categorical structures and their theories. Siberian Electronic Mathematical Reports, 2017, vol. 14, pp. 135–150. https://doi.org/10.17377/semi.2017.14.014
  27. Sudoplatov S.V. Relative 𝑒-spectra and relative closures for families of theories. Siberian Electronic Mathematical Reports, 2017, vol. 14, pp. 296–307. https://doi.org/10.17377/semi.2017.14.027
  28. Sudoplatov S.V. On semilattices and lattices for families of theories. Siberian Electronic Mathematical Reports, 2017, vol. 14, pp. 980–985. https://doi.org/10.17377/semi.2017.14.082
  29. Sudoplatov S.V. Classification of Countable Models of Complete Theories. Novosibirsk, NSTU Publ., 2018.
  30. Sudoplatov S.V. Combinations of structures. The Bulletin of Irkutsk State University. Series Mathematics, 2018, vol. 24, pp. 82–101. https://doi.org/10.26516/1997-7670.2018.24.82
  31. Sudoplatov S.V. Approximations of theories. Siberian Electronic Mathematical Reports, 2020, vol. 17, pp. 715–725. https://doi.org/10.33048/semi.2020.17.049
  32. Sudoplatov S.V. Hierarchy of families of theories and their rank characteristics. The Bulletin of Irkutsk State University. Series Mathematics, 2020, vol. 33, pp. 80–95. https://doi.org/10.26516/1997-7670.2020.33.80
  33. Sudoplatov S.V. Ranks for families of theories and their spectra. Lobachevskii Journal of Mathematics, 2021, vol. 42, no. 12, pp. 2959–2968. https://doi.org/10.1134/S1995080221120313
  34. Sudoplatov S.V. Arities and aritizabilities of first-order theories. Siberian Electronic Mathematical Reports, 2022, vol. 19, no. 2, pp. 889–901. https://doi.org/10.33048/semi.2022.19.075
  35. Tent K., Ziegler M. A Course in Model Theory. Cambridge, Cambridge University Press, 2012, 248 p.
  36. van den Dries L.P.D. Tame Topology and O-minimal Structures. Cambridge, Cambridge University Press, 1998, 182 p. https://doi.org/10.1017/CBO9780511525919

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