Abstract
In this article, we consider the notion of the Jonsson spectrum of some subclass of existentially closed models of a fixed Jonsson theory. For an arbitrary model of an arbitrary signature, the class of existentially closed models in that signature is considered, in which this arbitrary model is isomorphically embedded. The model-theoretical properties of the Kaiser Hull of this class are studied. In the language of central types of permissible enrichments, the properties of the considered fragments for special definable subsets of the semantic model of a fixed Jonsson theory are studied. The main result of this work is an estimate of the number of such fragments.
REFERENCES
J. Barwise, Handbook of Mathematical Logic, Model Theory, Part 1, Vol. 90 of Studies in Logic and the Foundations of Mathematics (North-Holland, Amsterdam, 1989).
B. Jonsson, ‘‘Homogeneous universal relational systems,’’ Math. Scand. 8, 137–142 (1960).
B. Jonsson, ‘‘Universal relational systems,’’ Math. Scand. 4, 193–208 (1956).
M. Morley and R. L. Vaught, ‘‘Homogeneous universal models,’’ Math. Scand. 11, 37–57 (1962).
T. G. Mustafin, ‘‘Generalized Jonsson conditions and a description of generalized Jonsson theories of boolean algebras,’’ Sib. Adv. Math. 10 (3), 1–58 (2000).
A. R. Yeshkeyev, ‘‘Perfect Jonsson theories,’’ in Proceedings of the 3rd International Conference on Algebra (Krasnoyarsk, 1993).
A. R. Yeshkeyev, ‘‘On Jonsson stability and some of its generalizations,’’ J. Math. Sci. 166, 646–654 (2010).
A. R. Yeshkeyev, ‘‘The structure of lattices of positive existential formulae of (\(\triangle\)-PJ)-theories,’’ Sci. Asia, J. Sci. Soc. Thailand 39, 19–24 (2013).
A. R. Yeshkeyev, ‘‘Companions of the fragments in the Jonsson sets,’’ Bull. Karaganda Univ.: Math. 85, 41–45 (2017).
B. Poizat and A. R. Yeshkeyev, ‘‘Positive Jonsson theories,’’ Logica Univ. 12, 101–127 (2018).
A. R. Yeshkeyev and M. T. Omarova, ‘‘Companions of \((n_{1},n_{2})\)-Jonsson theory,’’ Bull. Karaganda Univ.: Math. 96 (4), 75–80 (2019).
Y. T. Mustafin, ‘‘Quelques proprietes des theories de Jonsson,’’ J. Symbol. Logic 67, 528–536 (2002).
A. R. Yeshkeyev, ‘‘On \(J\)-stability of Jonsson’s theories,’’ in Proceedings of the 9th Asian Logic Conference (2005), pp. 73–74.
A. R. Yeshkeyev, ‘‘Properties of companions of Jonsson’s theory,’’ in Proceedings of the Model Theory and Algebra France–Kazakhstan Conference (Astana, 2005), p. 77.
A. R. Yeshkeyev, M. T. Kassymetova, and N. K. Shamatayeva, ‘‘Model-theoretic properties of the \(\#\)-companion of a Jonsson set,’’ Euras. Math. J. 9 (2), 68–81 (2018).
A. R. Yeshkeyev and M. T. Kassymetova, Jonsson Theories and their Model Classes (KarGU, Karaganda, 2016) [in Russian].
T. G. Mustafin, ‘‘On similarities of complete theories,’’ in Proceedings of the Logic Colloquium ’90: Annual European Summer Meeting of the Association for Symbolic Logic (Helsinki, 1990), pp. 259–265.
A. R. Yeshkeyev and O. I. Ulbrikht, ‘‘JSp-cosemanticity and JSB property of abelian groups,’’ Sib. Elektron. Mat. Izv. 13, 861–874 (2016).
A. R. Yeshkeyev and O. I. Ulbrikht, ‘‘JSp-cosemanticity of \(R\)-modules,’’ Sib. Elektron. Mat. Izv. 16, 1233–1244 (2019).
A. R. Yeshkeyev, M. T. Kassymetova, and O. I. Ulbrikht, ‘‘Independence and simplicity in Jonsson’s theories with abstract geometry,’’ Sib. Elektron. Mat. Izv. 16, 433–455 (2019).
D. W. Kueker, ‘‘Core structures for theories,’’ Fundam. Math. 89, 154–171 (1973).
T. G. Mustafin, ‘‘On a strong base of elementary types of theories,’’ Sib. Mat. Zh. 18, 1356–1366 (1977).
T. G. Mustafin, ‘‘New notions of stability of theories,’’ in Model Theory, Proceedings of the Sov.-Fr. Colloquium, Karaganda, USSR (1990), pp. 112–125.
E. A. Palyutin, ‘‘\(E^{*}\)-stable theories,’’ Algebra Logic 42, 112–120 (2003).
W. Hodges, Encyclopedia of Mathematics and its Applications, Vol. 42: Model Theory, Ed. by G.-C. Rota (Cambridge Univ. Press, Cambridge, 1993).
A. R. Yeshkeyev, M. T. Omarova, and G. E. Zhumabekova, ‘‘The \(J\)-minimal sets in the hereditary theories,’’ Bull. Karaganda Univ.: Math. 94 (2), 92–98 (2019).
A. R. Yeshkeyev and M. T. Kassymetova, ‘‘Pregeometry on the subsets of Jonsson theory’s semantic model,’’ Bull. Karaganda Univ.: Math. 90 (2), 88–92 (2018).
A. R. Yeshkeyev and M. T. Omarova, ‘‘An essential base of the central types of the convex theory,’’ Bull. Karaganda Univ.: Math. 101 (1), 119–126 (2021).
J. T. Baldwin and D. W. Kueker, ‘‘Algebraically prime models,’’ Ann. Math. Logic 20, 289–330 (1981).
R. L. Vaught, ‘‘Denumerable models of complete theories,’’ in Proceedings of the Symposium on Foundations of Mathematics, Infinitistic Methods (Warsaw, 1959).
A. Robinson, Introduction to the Model Theory and the Metamathematics of Algebra (North-Holland, Amsterdam, 1963).
A. R. Yeshkeyev, ‘‘Model-theoretical questions of the Jonsson spectrum,’’ Bull. Karaganda Univ.: Math. 98 (2), 165–173 (2020).
J. T. Baldwin, Categoricity (Univ. Illinois, Chicago, 1988).
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This work was supported by the Science Committee of the Ministry of Education and Science of the Republic of Kazakhstan (grant AP09260237).
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Yeshkeyev, A.R., Ulbrikht, O.I. & Omarova, M.T. The Number of Fragments of the Perfect Class of the Jonsson Spectrum. Lobachevskii J Math 43, 3658–3673 (2022). https://doi.org/10.1134/S199508022215029X
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DOI: https://doi.org/10.1134/S199508022215029X