Model-theoretic properties of J-superstable Jonsson theories in classes defined by cosemanticness
DOI:
https://doi.org/10.31489/2026m1/238-247Keywords:
Jonsson theory, semantic model, perfect Jonsson theory, hereditary Jonsson theory, permissible enrichment, J-λ-stable theory, J-superstable theory, J-P-superstable theory, J-orthogonal type, J-nonmultidimensional theoryAbstract
This article deals with the problems of model-theoretic characterization of J-superstable Jonsson theories. The characteristic features of such theories are analyzed in terms of J-stability, J-P-superstability, and J-nonmultidimensionality. The need to generalize classical stability notions to the framework of Jonsson theories is identified and justified. The concepts of J-stationary and J-orthogonal types are introduced, and their role in describing the dimensional structure of existentially closed models is examined. It is shown that every JFλα-saturated model embeds as a submodel of the semantic model of a Jonsson theory T. Based on the notion of J-stationarity, a theory of independence for existential types is developed, and the notions of J-basis and J-dimension are defined. The equivalence between J-nonmultidimensionality, J-P-superstability, and J-P-stability is established, providing precise criteria for the model-theoretic classification of Jonsson theories. The results contribute to the refinement of model-theoretic tools for analyzing stability and dimensionality within the framework of Jonsson theories and place these findings in the broader context of modern classification theory, highlighting Jonsson theories as a natural generalization of elementary theories. These results clarify the interaction between saturation, independence, and dimensionality in Jonsson theories and provide a unified framework for further developments in their model-theoretic classification.
References
Poizat, B. (1983). Paires de structure stables. The Journal of Symbolic Logic, 48(2), 239–249. https://doi.org/10.2307/2273543
Bouscaren, E. (1989). Elementary pairs of models. Annals of Pure and Appl. Logic, 45(2), 129–137. https://doi.org/10.1016/0168-0072(89)90057-2
Palyutin, E.A. (2003). E∗-stabilnye teorii [E∗-stable theories]. Algebra i logika — Algebra and logic, 42(2), 194–210 [in Russian].
Mustafin, T.G., & Nurmagambetov, T.A. (1990). O P-stabilnosti polnykh teorii [On P-stability of complete theories]. Strukturnye svoistva algebraicheskikh sistem — Structural properties of algebraic systems, 88–100 [in Russian].
Yeshkeyev, A.R., Zhumabekova, G.E., & Tungushbayeva, I.O. (2023). The central type of a semantic pair. Logic Colloquium European Summer Meeting of the Association for Symbolic Logic: Collection of Abstracts, 184. Italy: University of Milan.
Kassymetova, M.T., & Zhumabekova, G.E. (2024). Svoistva nemultirazmernosti dlia klassa semanticheskikh par [Nonmultidimensional properties for the class of semantic pairs]. Traditsionnaia mezhdunarodnaia aprelskaia matematicheskaia konferentsiia v chest Dnia uchenogo Respubliki Kazakhstan — Traditional International April Mathematical Conference in honor of the Day of Scientists of the Republic of Kazakhstan: Collection of Abstracts, 222. Almaty: Institute of Mathematics and Mathematical Modeling [in Russian].
Amanbekov, S.M., Onerkhaan A., & Tungushbayeva, I.O. (2025). The properties of amalgamation and joint embedding in the meaning of positive Jonsson theories. Kazakh Mathematical Journal, 25(2), 6–18. https://doi.org/10.70474/18bp7004
Yarullina, A.R., & Kaliolla, D.Ye. (2022). The perfect Jonsson S-acts. Proceedings of the International Scientific Conference “Actual Problems of Mathematics, Mechanics and Informatics” dedicated to the 80th anniversary of Professor T.G. Mustafin, 40–42. Karaganda: Karaganda Buketov University.
Omarova, M.T., & Amanbekov, S.M. (2024). 1-perfectness of fragments of normal Jonsson theory. International Conference Mal’tsev Meeting: Collection of Abstracts, 243. Novosibirsk: Sobolev Institute of Mathematics, Novosibirsk State University.
Popova, N.V., & Amanbekov, S.M. (2024). The properties of 1-model companions of Jonsson sets in normal Jonsson theory. International Conference Mal’tsev Meeting: Collection of Abstracts, 244. Novosibirsk: Sobolev Institute of Mathematics, Novosibirsk State University.
Kassymetova, M.T., Tungushbayeva, I.O., & Yarullina, A.R. (2025). Syntactic similarity of Hybrids of classes of normal Jonsson theories. International Conference Actual problems of analysis, differential equations and algebra (EMJ–2025) dedicated to the 15th anniversary of the Eurasian Mathematical Journal: Collection of Abstracts, 162. Astana: L.N. Gumilyov Eurasian National University.
Kassymetova, M.T., Mussina, N.M., & Amanbekov, S.M. (2021). Companions of hybrids of fragments of theoretical subsets. International Conference Mal’tsev Meeting: Collection of Abstracts, 167. Novosibirsk: Sobolev Institute of Mathematics, Novosibirsk State University.
Tungushbayeva, I.O., & Ulbrikht, O.I. (2024). On theories with amalgamation and joint embedding properties. International Conference Mal’tsev Meeting: Collection of Abstracts, 171. Novosibirsk: Sobolev Institute of Mathematics, Novosibirsk State University.
Barwise, J. (1977). Handbook of Mathematical Logic. Studies in Logic and the Foundations of Mathematics (Vol. 90). Amsterdam: North-Holland.
Yeshkeyev, A.R. (2024). Teorii i ikh modeli [Theories and their models]: monograph (Vols. 1–2). Karaganda: Izdatelstvo KarU [in Russian].
