Abstract
We describe the algebras of binary formulas for countably categorical weakly circularly minimal theories with 1-transitive nonprimitive automorphism group and trivial definable closure having convexity rank 1. We find some criterion for commutativity of the algebras.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Sudoplatov S.V., Classification of Countable Models of Complete Theories. Parts 1 and 2, NGTU, Novosibirsk (2018) [Russian].
Shulepov I.V. and Sudoplatov S.V., “Algebras of distributions for isolating formulas of a complete theory,” Sib. Electr. Math. Reports, vol. 11, 380–407 (2014).
Kulpeshov B.Sh. and Sudoplatov S.V., “On algebras of distributions of binary isolating formulas for quite o-minimal theories,” Izv. NAS RK. Ser. Fiz.-Mat., vol. 300, no. 2, 5–13 (2015).
Emel’yanov D.Yu., Kulpeshov B.Sh., and Sudoplatov S.V., “Algebras of distributions for binary formulas in countably categorical weakly o-minimal structures,” Algebra Logic, vol. 56, no. 1, 13–36 (2017).
Baikalova K.A., Emel’yanov D.Yu., Kulpeshov B.Sh., Palyutin E.A., and Sudoplatov S.V., “On algebras of distributions of binary isolating formulas for theories of abelian groups and their ordered enrichments,” Russian Math. (Iz. VUZ), vol. 64, no. 4, 1–12 (2018).
Emel’yanov D.Yu., Kulpeshov B.Sh., and Sudoplatov S.V., “Algebras of distributions of binary isolating formulas for quite o-minimal theories,” Algebra Logic, vol. 57, no. 6, 429–444 (2018).
Emel’yanov D.Yu., Kulpeshov B.Sh., and Sudoplatov S.V., “Algebras of binary formulas for compositions of theories,” Algebra Logic, vol. 59, no. 4, 295–312 (2020).
Kulpeshov B.Sh. and Sudoplatov S.V., “Algebras of binary formulas for weakly circularly minimal theories with non-trivial definable closure,” Lobachevskii J. Math., vol. 43, no. 12, 3532–3540 (2022).
Altaeva A.B., Kulpeshov B.Sh., and Sudoplatov S.V., “Algebras of distributions of binary isolating formulas for almost \( \omega \)-categorical weakly o-minimal theories,” Algebra Logic, vol. 60, no. 4, 241–262 (2021).
Bhattacharjee M., Macpherson H.D., Möller R. G., and Neumann P.M., Notes on Infinite Permutation Groups, Springer, Berlin (1998) (Lect. Notes Math.; vol. 1698).
Cameron P.J., “Orbits of permutation groups on unordered sets. II,” J. London Math. Soc., vol. 2, 249–264 (1981).
Droste M., Giraudet M., Macpherson H.D., and Sauer N., “Set-homogeneous graphs,” J. Combinat. Theory. Ser. B, vol. 62, no. 1, 63–95 (1994).
Kulpeshov B.Sh. and Macpherson H.D., “Minimality conditions on circularly ordered structures,” Math. Logic Quart., vol. 51, no. 4, 377–399 (2005).
Kulpeshov B.Sh., “On \( \aleph_{0} \)-categorical weakly circularly minimal structures,” Math. Logic Quart., vol. 52, no. 6, 555–574 (2006).
Kulpeshov B.Sh., “Definable functions in the \( \aleph_{0} \)-categorical weakly circularly minimal structures,” Sib. Math. J., vol. 50, no. 2, 282–301 (2009).
Kulpeshov B.Sh. and Verbovskiy V.V., “On weakly circularly minimal groups,” Math. Logic Quart., vol. 61, no. 1, 82–90 (2015).
Kulpeshov B.Sh., “On indiscernibility of a set in circularly ordered structures,” Sib. Electr. Math. Reports, vol. 12, 255–266 (2015).
Kulpeshov B.Sh. and Altaeva A.B., “Binary formulas in countably categorical weakly circularly minimal structures,” Algebra Logic, vol. 55, no. 3, 226–241 (2016).
Kulpeshov B.Sh., “On almost binarity in weakly circularly minimal structures,” Eurasian Math. J., vol. 7, no. 2, 38–49 (2016).
Altaeva A.B. and Kulpeshov B.Sh., “Almost binarity of countably categorical weakly circularly minimal structures,” Math. Notes, vol. 110, no. 6, 813–829 (2021).
Macpherson H.D., Marker D., and Steinhorn C., “Weakly o-minimal structures and real closed fields,” Trans. Amer. Math. Soc., vol. 352, no. 12, 5435–5483 (2000).
Kulpeshov B.Sh. and Altaeva A.B., “Equivalence-generating formulas in weakly circularly minimal structures,” Reports of the National Academy of Sciences of the Republic of Kazakhstan, vol. 2, 5–10 (2014).
Funding
This research has been funded by the Science Committee of the Ministry of Science and Higher Education of the Republic of Kazakhstan (Grant AP19674850).
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
As author of this work, I declare that I have no conflicts of interest.
Additional information
Translated from Sibirskii Matematicheskii Zhurnal, 2024, Vol. 65, No. 2, pp. 318–331. https://doi.org/10.33048/smzh.2024.65.207
Publisher's Note
Pleiades Publishing remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Kulpeshov, B.S. Algebras of Binary Formulas for Weakly Circularly Minimal Theories with Trivial Definable Closure. Sib Math J 65, 316–327 (2024). https://doi.org/10.1134/S0037446624020071
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0037446624020071
Keywords
- algebra of binary formulas
- countably categorical theory
- weak circular minimality
- circularly ordered structure