E -combinations of ℵ 0 -categorical linear orderings

  • A. B. Altayeva Al-Farabi Kazakh National University, Institute of Mathematics and Mathematical Modeling of MES RK
  • B. Sh Kulpeshov Kazakh-British Technical University, Institute of Mathematics and Mathematical Modeling of MES RK
  • S. V. Sudoplatov Sudoplatov S.V., Sobolev Institute of Mathematics, Novosibirsk State Technical University

Abstract

In a series of S.V. Sudoplatov’s works the topological properties of families of theories arestudied. He introduced the concepts of the P-operator and the E-operator, allowing to studythe connections between theories regarding suitable closure operators. These operators make itpossible to generate new theories by means of the considered families of theories, and also find insome cases minimal or smallest generating sets. This research was continued in collaborative workof B.Sh. Kulpeshov and S.V. Sudoplatov for families of ordered theories, including families ofquite o-minimal theories. This article explores the E-combinations of countably many countablycategorical linearly ordered structures of pure linear order. Criterion for countable categoricityof E-combination of countably many copies of an arbitrary countably categorical linear order isobtained. As a consequence, a description of the countable spectrum of such a combination isobtained.

References

[1] Sudoplatov S.V., ”Combinations of structures”, The Bulletin of Irkutsk State University. Series “Mathematics”, vol. 24 2018): 82-101.
[2] Sudoplatov S.V., ”Closures and generating sets related to combinations of structures”, The Bulletin of Irkutsk State niversity. Series “Mathematics”, vol. 16 (2016): 131-144.
[3] Sudoplatov S.V., ”Families of language uniform theories and their generating sets”, The Bulletin of Irkutsk State University. Series “Mathematics”, vol. 17 (2016): 62-76.
[4] Sudoplatov S.V., ”On semilattices and lattices for families of theories”, Siberian Electronic Mathematical Reports, vol. 14 (2017): 980-985.
[5] Sudoplatov S.V., ”Combinations related to classes of finite and countably categorical structures and their theories”, Siberian Electronic Mathematical Reports, vol. 14 (2017): 135-150.
[6] Sudoplatov S.V., ”Relative e-spectra, relative closures, and semilattices for families of theories”, Siberian Electronic Mathematical Reports, vol. 14 (2017): 296-307.
[7] Pavlyuk In.I., Sudoplatov S.V., ”Families of theories of Ab elian groups and their closures”, Bulletin of Karaganda University. Mathematics, vol. 92, No. 4 (2018): 72-78.
[8] Kulp eshov B.Sh., Sudoplatov S.V., ”On P-combinations of ordered theories”, Traditional International April Mathematical Conference (abstracts), Institute of Mathematics and Mathematical Modeling, Almaty (2019): 30-31.
[9] Kulpeshov B.Sh., Sudoplatov S.V., ”On Ehrehfeuchtness of a P-combination of irdered theories”, Materials of the International Conference ”Algebra and Mathematical Logic: theory and applications”, Kazan: Kazan Federal University (2019): 131-133.
[10] Kulpeshov B.Sh., Sudoplatov S.V., ”P-combination of ordered structures”, Abstracts of the International Conference ”Maltsev Meeting”, Novosibirsk: Sobolev Institute of Mathematics (2019): 190.
[11] Rosenstein J.G., ” ℵ 0-categoricity of linear orderings”, Fundamenta Mathematicae, vol. 64 (1969): 1-5.
[12] Feferman S., Vaught R., ”The first order properties of products of algebraic systems”, Fundamenta Mathematicae, vol. 47 1959): 57-103.
[13] Kulpeshov B.Sh., ”Convexity rank and orthogonality in weakly o-minimal theories”, News of National Academy of Sciences of the Republic of Kazakhstan, series physics-mathematics, vol. 227 (2003): 26-31.
[14] Kulpeshov B.Sh., Sudoplatov S.V., ”Vaught’s conjecture for quite o-minimal theories”, Annals of Pure and Applied Logic, vol. 168 (2017): 129-149.
[15] Kulpeshov B.Sh., ”Countably categorical quite o-minimal theories”, Journal of Mathematical Sciences, vol. 188 (2013): 387-397.
[16] Rubin M., ”Theories of linear order”, Israel Journal of Mathematics, vol. 17 (1974): 392-443.
[17] Sudoplatov S.V., ”Classification of countable models of complete theories”, Part 1, Novosibirsk: Novosibirsk State Technical University Publishing House (2018): 376 c.
[18] Kulpeshov B.Sh., Sudoplatov S.V., ”Linearly ordered theories which are nearly countably categorical”, Mathematical Notes, vol. 101 (2017): 475-483.
[19] Kulpeshov B.Sh., ”Weakly o-minimal structures and some of their properties”, The Journal of Symbolic Logic, vol. 63 (1998): 1511-1528.
[20] Mayer L.L., ”Vaught’s conjecture for o-minimal theories”, The Journal of Symbolic Logic, vol. 53 (1988): 146-159.
Published
2019-10-27
How to Cite
ALTAYEVA, A. B.; KULPESHOV, B. Sh; SUDOPLATOV, S. V.. E -combinations of ℵ 0 -categorical linear orderings. Journal of Mathematics, Mechanics and Computer Science, [S.l.], v. 103, n. 3, p. 3-12, oct. 2019. ISSN 2617-4871. Available at: <https://bm.kaznu.kz/index.php/kaznu/article/view/655>. Date accessed: 22 oct. 2020. doi: https://doi.org/10.26577/JMMCS-2019-3-21.
Keywords Е-combination, ordered theory, linearly ordered structure, countable categoricity, countable spectrum