Normality and superperfection of Jonsson spectra
DOI:
https://doi.org/10.31489/2026m3/200-209Keywords:
Jonsson spectrum, Jonsson theory, Kaiser class, superperfection, K_T-normality, cl-normality, Jonsson pregeometry, central types, P-stability, beautiful pairsAbstract
This paper investigates the structural properties of Jonsson spectra with respect to the notions of KTnormality, cl-normality, and superperfection. Particular attention is paid to the role of the Kaiser class KT as an internal structural layer of a Jonsson spectrum. In contrast to the classical approach based exclusively on the class of existentially closed models, a more general normalization framework for Jonsson spectra is developed through the consistency of Jonsson pregeometry with the Kaiser class. The study is carried out within the framework of Jonsson model theory and relies on methods of pregeometry and spectral analysis. The characteristic features of Jonsson spectra are examined from the standpoint of their interaction with structural closure operators and the Kaiser class. The notions of a KT-normal Jonsson spectrum, a cl-normal Jonsson spectrum, an ideal Jonsson spectrum, and a KT-beautiful pair are introduced and formally characterized. Criteria for the stability of Jonsson spectra in the superperfect case are obtained. Connections between spectrum normality, Jonsson pregeometry, central types, the heredity of Jonsson theories, and P-stability are established. The results contribute to the further development of the structural theory of Jonsson spectra and provide new tools for their investigation.
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 Applied 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. Sbornik nauchnykh trudov — Structural properties of algebraic systems. Collected scientific papers (pp. 88–100). Karaganda [in Russian].
Yeshkeyev, A.R. (2024). Teorii i ikh modeli [Theories and their models]: monograph (Vols. 1–2). Karaganda: Izdatelstvo KarU [in Russian].
Yeshkeyev, A.R., Kasymetova, M.T., & Shamatayeva, N.K. (2018). Model-theoretic properties of the ]-companion of a Jonsson set. Eurasian Mathematical Journal, 9(2), 68–81. https://doi.org/10.32523/2077-9879-2018-9-2-68-81
Yeshkeyev, A.R., Tungushbayeva, I.O., & Ulbrikht, O.I. (2024). On some types of algebras of a Jonsson spectrum. Siberian Electronic Mathematical Reports, 21(2), 866–881. https://doi.org/10.33048/semi.2024.21.057
Yeshkeyev, A.R., Ulbrikht, O.I., & Omarova, M.T. (2022). The number of fragments of the perfect class of the Jonsson spectrum. Lobachevskii Journal of Mathematics, 43, 3658–3673. https://doi.org/10.1134/S199508022215029X
Omarova, M.T., & Amanbekov, S.M. (2024). 1-perfectness of fragments of normal Jonsson theory. International Conference Mal’tsev Meeting (September 20–24, 2021): Collection of Abstracts (p. 243). Novosibirsk: Sobolev Institute of Mathematics, Novosibirsk State University.
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
Poizat, B., & Yeshkeyev, A.R. (2023). Back and Forth in Positive Logic. Logic in Question. Studies in Universal Logic, 603–609. https://doi.org/10.1007/978-3-030-94452-0_31
Yeshkeyev, A.R., Tungushbayeva, I.O., & Koshekova, A.K. (2024). The cosemanticness of Kaiser hulls of fixed classes of models. Bulletin of the Karaganda University. Mathematics Series, 1(113), 208–217. https://doi.org/10.31489/2024m1/208-217
Yeshkeyev, A.R., Tungushbayeva, I.O., & Kassymetova, M.T. (2025). On the Kaiser Class of a Jonsson Theory. Symmetry, 17(6), 870. https://doi.org/10.3390/sym17060870
Zhumabekova, G.E. (2023). Model-theoretic properties of semantic pairs and e.f.c.p. in Jonsson spectrum. Bulletin of the Karaganda University. Mathematics Series, 4(112), 185–193. https://doi.org/10.31489/2023m4/185-193
Yeshkeyev, A.R., Yarullina, A.R, & Kassymetova, M.T. (2024). Jonsson existentially closed unars of expanded signature. Bulletin of Abai KazNPU. Series of Physical and Mathematical Sciences, 3(87), 17–26. https://doi.org/10.51889/2959-5894.2024.87.3.002
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 (September 20–24, 2021): Collection of Abstracts (p.244). Novosibirsk: Sobolev Institute of Mathematics, Novosibirsk State University.
Kassymetova, M.T., Mussina, N.M., & Amanbekov, S.M. (2021). Companions of hybrids of fragments of theoretical subsets. International Conference Mal’tsev Meeting (September 20–24, 2021): Collection of Abstracts(p.167). 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. (January 7–11, 2025): Collection of Abstracts (p.162). Astana: L.N. Gumilyov Eurasian National University.
Barwise, J. (1977). Handbook of Mathematical Logic. Studies in Logic and the Foundations of Mathematics: Volume 90. North-Holland, Amsterdam/New York/Oxford.
Mustafin, Y. (2002). Quelques proprietes des Theories de Jonsson. Journal of Symbolic Logic, 67(2), 528–536. https://doi.org/10.2178/jsl/1190150095
