рус eng
«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». 2025. Vol 53

Non-Orthogonality of 1-types in Theories with a Linear Order

Author(s)

Bektur Baizhanov1

Olzhas Umbetbayev1,2

Tatyana Zambarnaya1

1 Institute of Mathematics and Mathematical Modeling, Almaty, Kazakhstan 

2 Kazakh-British Technical University, Almaty, Kazakhstan

Abstract
Non-orthogonality of complete types is an important concept for such classes of first-order theories as o-minimal, weakly-o-minimal and quite o-minimal theories. This concept is used in studying countable spectrum of such theories, since orthogonality affects omission and realization of types. Further study of the Vaught’s conjecture for small ordered theories requires the use of the relation between incomplete types, in particular, convex closures of 1-types. In this paper, two notions of non-orthogonality of convex incomplete types are introduced. Connections between different kinds of non-orthogonality are shown. Theorems on preservation of properties of types under non-orthogonality are proven.
About the Authors

Bektur Baizhanov, Dr. Sci. (Phys.–Math.), Prof., Corresponding Member of NAS RK, Institute of Mathematics and Mathematical Modeling, Almaty, 050010, Kazakhstan, baizhanov@math.kz, https://orcid.org/0000-0002-3743-7404 

Olzhas Umbetbayev, Master of mathematics, Researcher, Institute of Mathematics and Mathematical Modeling, Almaty, 050010, Kazakhstan, umbetbayev@math.kz, https://orcid.org/0000-0002-0211-2159 

Tatyana Zambarnaya, PhD, Leading Researcher, Institute of Mathematics and Mathematical Modeling, Almaty, 050010, Kazakhstan, zambarnaya@math.kz, https://orcid.org/0000-0001-7203-170

For citation

Baizhanov B., Umbetbayev O., Zambarnaya T. Non-Orthogonality of 1- types in Theories with a Linear Order. The Bulletin of Irkutsk State University. Series Mathematics, 2025, vol. 53, pp. 131–140. https://doi.org/10.26516/1997-7670.2025.53.131

Keywords
UDC
510.67
MSC
03C64
DOI
https://doi.org/10.26516/1997-7670.2025.53.131
References
  1. Alibek A., Baizhanov B.S., Kulpeshov B.Sh., Zambarnaya T.S. Vaught’s conjecture for weakly o-minimal theories of convexity rank 1. Annals of Pure and Applied Logic, 2018, vol. 169, pp. 1190–1209. https://doi.org/10.1016/j.apal.2018.06.003 
  2. 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, pp. 1382–1414. https://doi.org/10.2307/2695114 
  3. Baizhanov B.S. Orthogonality of one-types in weakly o-minimal theories. Algebra and Model Theory II, 1999, pp. 3–28. 
  4. Baizhanov B.S., Baizhanov S.S., Saulebayeva T., Zambarnaya T.S. One-formulas and one-types in ordered theories. Matematicheskii zhurnal [Mathematical Journal], 2016, vol. 16, pp. 104–125. 
  5. Baizhanov B.S., Tazabekova N.S., Yershigeshova A.D., Zambarnaya T.S. Types in small theories. Matematicheskii zhurnal [Mathematical Journal], 2015, vol. 15, pp. 38–56. 
  6. Baizhanov B., Umbetbayev O. Constant expansion of theories and the number of countable models. Siberian Electronic Mathematical Reports, 2023, vol. 20, no. 2, pp. 1037–1051. https://doi.org/10.33048/semi.2023.20.064 
  7. Baldwin J.T., Lachlan A.H. On strongly minimal sets. Journal of Symbolic Logic, 1971, vol. 36, pp. 79–96. 
  8. Benda M. Remarks on countable models. Fund. Math. 1974, pp. 107–119. 
  9. Casanovas E. The number of countable models. Model Theory Seminar. University of Barcelona, 2002, 24 p. 
  10. Hart B., Hrushovski E., Laskowski M.C. The uncountable spectra of countable theories. Annals of Mathematics, 2000, vol. 152, no. 1, pp. 207–257. 
  11. Kulpeshov B.Sh. Maximality of the countable spectrum in small quite o-minimal theories. Algebra and Logic, 2019, vol. 58, no. 2, pp. 137—143. 
  12. Kulpeshov B.Sh. Vaught’s conjecture for weakly o-minimal theories of finite convexity rank. Izvestiya: Mathematics, 2020, vol. 84, no. 2, pp. 324–347. https://doi.org/10.1070/IM8894
  13. Kulpeshov B.Sh., Sudoplatov S.V. Linearly ordered theories which are nearly countably categorical. Mathematical Notes, 2017, vol. 101, no. 3, pp. 475-–483. 
  14. Kulpeshov B.Sh., Sudoplatov S.V. Vaught’s conjecture for quite o-minimal theories. Annals of Pure and Applied Logic, 2017, vol. 168, pp. 129–149. https://doi.org/10.1016/j.apal.2016.09.002  
  15. Mayer L. Vaught’s conjecture for o-minimal theories. Journal of Symbolic Logic, 1988, vol. 53, no. 1, pp. 146–159. 
  16. Moconja S., Tanovic P. Stationarily ordered types and the number of countable models. Annals of Pure and Applied Logic, 2020, vol. 171, no. 3, article number 102765. 
  17. Morley M.D. The number of countable models. Journal of Symbolic Logic, 1970, vol. 35, pp. 14–18. 
  18. Pillay A. Number of countable models. Journal of Symbolic Logic, 1978, vol. 43, no 3. pp. 492–496. 
  19. Rubin M. Theories of linear order. Israel Journal of Mathematics, 1974, vol. 17, pp. 392–443. 
  20. Shelah S. Classification theory and the number of non-isomorphic models. Amsterdam, North-Holland, 1990, 705 p. https://doi.org/10.2307/2275113 
  21. Shelah S., Harrington L., Makkai M. A proof of Vaught’s conjecture for 𝜔-stable theories. Israel Journal of Mathematics, 1984, vol. 49, pp. 259–280. 
  22. Sudoplatov S.V. Classification of countable models of complete theories. Novosibirsk, NSTU Publ., 2018. 
  23. Vaught R. Denumerable models of complete theories. Infinistic Methods. London, Pergamon, 1961, pp. 303–321. 
  24. Woodrow R. Theories with a finite number of counttable models. Journal of Symbolic Logic, 1978, vol. 43, no. 3, pp. 442-–455

Full text (english)