Geodesic
Independence and simplicity in Jonsson theories with abstract geometry
Sibirskie èlektronnye matematičeskie izvestiâ, Tome 18 (2021) no. 1, pp. 433-455.

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The concepts of forking and independence are examined in the framework of the study of Jonsson theories and the fixed Jonsson spectrum. The axiomatically given property of nonforking satisfies the classical notion of nonforking in the sense of S. Shelah and the approach to this concept by Laskar-Poizat. On this basis, the simplicity of the Jonsson theory is determined and the Jonsson analog of the Kim-Pillay theorem is given. Abstract pregeometry on definable subsets of the Jonsson theory's semantic model is defined. The properties of Morley rank and degree for definable subsets of the semantic model are considered. A criterion of uncountable categoricity for the hereditary Jonsson theory in the language of central types is proved.
Keywords: Jonsson theory, existentially closed model, Morley rank, Jonsson spectrum, Jonsson set, a fragment of Jonsson set, Jonsson independence, Jonsson nonforking, Jonsson simplicity, strong minimality, pregeometry, modular geometry.
Mots-clés : cosemanticness, central type 
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A. R. Yeshkeyev; M. T. Kassymetova; O. I. Ulbrikht. Independence and simplicity in Jonsson theories with abstract geometry. Sibirskie èlektronnye matematičeskie izvestiâ, Tome 18 (2021) no. 1, pp. 433-455. http://geodesic.mathdoc.fr/item/SEMR_2021_18_1_a7/

[1] T.G. Mustafin, “New notions of stability of theories”, Model theory, Proc. Sov.-Fr. Colloq. (Karaganda/USSR, 1990), 112–125 | MR | Zbl

[2] E.A. Palyutin, “$E^*$-stable theories”, Algebra Logic, 42:2 (2003), 112–120 | DOI | MR | Zbl

[3] A.R. Yeshkeyev, O.I. Ulbrikht, “$Jsp$-cosemanticness and JSB property of abelian groups”, Sib. Électron. Math. Izv., 13 (2016), 861–874 | MR | Zbl

[4] A.R. Yeshkeyev, O.I. Ulbrikht, “$JSp$-cosemanticness of $R$-modules”, Sib. Électron. Math. Izv., 16 (2019), 1233–1244 | DOI | MR | Zbl

[5] A.R. Yeshkeyev, M.T. Kassymetova, N.K. Shamatayeva, “Model-theoretic properties of the $\sharp$-companion of a Jonsson set”, Eurasian mathematical journal, 9:2 (2018), 68–81 | DOI | MR | Zbl

[6] A.R. Yeshkeyev, “The structure of lattices of positive existential formulae of $\Delta$-$PJ$-theories”, ScienceAsia, 39 (2013), 19–24 | DOI

[7] B. Poizat, A. Yeshkeyev, “Positive Jonsson theories”, Log. Univers., 12:1-2 (2018), 101–127 | DOI | MR | Zbl

[8] S. Shelah, Classification theory and the number of non-isomorphic models, Studies in Logic and the Foundations of Mathematics, 92, North-Holland, Amsterdam etc., 1990 | MR | Zbl

[9] A. Marcja, C. Toffalori, A guide to classical and modern model theory, Trends in Logic-Studia Logica Library, 19, Springer, Dordrecht, 2003 | MR | Zbl

[10] J. Barwise (ed.), Handbook of mathematical logic, Studies in Logic and the Foundations of Mathematics, 90, North-Holland Publishing Company, Amsterdam etc., 1982 | MR | Zbl

[11] G.E. Sacks, Saturated model theory, Mir, M., 1976 ; 2010 | MR | Zbl

[12] Y.T. Mustafin, “Quelques proprietes des theories de Jonsson”, J. Symb. Log., 67:2 (2002), 528–536 | DOI | MR | Zbl

[13] A.R. Yeshkeyev, M.T. Kassymetova, Jonsson theories and their classes of models, KarGU, Karaganda, 2009

[14] D. Lascar, B. Poizat, “An introduction to forking”, J. Symb. Log., 44:3 (1979), 330–350 | DOI | MR | Zbl

[15] J.T. Baldwin, Fundamentals of stability theory, Springer, Berlin etc, 1988 | MR | Zbl

[16] A.R. Yeshkeyev, “Model-theoretic properties of Jonsson fragments”, Bulletin of the Karaganda University. Mathematics Series, 4(76) (2014), 37–41

[17] A.R. Yeshkeyev, M.T. Kasymetova, “Properties of lattices of the existential formulas of Jonsson fragments”, Bulletin of the Karaganda University. Mathematics Series, 3(79) (2015), 25–32

[18] A.R. Yeshkeyev, “Convex fragmens of strongly minimal Jonsson sets”, Bulletin of the Karaganda University. Mathematics Series, 1(77) (2015), 67–72 http://rep.ksu.kz:80//handle/data/4094

[19] A.R. Yeshkeyev, “Companions of the fragments in the Jonsson enrichment”, Bulletin of the Karaganda University. Mathematics Series, 1(85) (2017), 41–45 | DOI

[20] F.O. Wagner, Simple Theories, Mathematics and Its Applications, 503, Kluwer Academic Publishers, Dordrecht, 2000 | MR | Zbl

[21] C.C. Chang, H.J. Keisler, Model Theory, Studies in Logic and the Foundations of Mathematics, 73, North-Holland, Amsterdam etc., 1990 | MR | Zbl

[22] T.G. Mustafin, Stable theories, Karaganda, 1981 | Zbl

[23] T.G. Mustafin, The number of models of theories, Karaganda, 1983

[24] A.R. Yeshkeyev, “On Jonsson stability and some of its generalizations”, J. Math. Sci., 166:5 (2010), 646–654 | DOI | MR | Zbl

[25] E. Hrushovski, B. Zilber, “Zariski geometries”, Bull. Am. Math. Soc., New Ser., 28:2 (1993), 315–323 | DOI | MR | Zbl

[26] D. Marker, Model Theory: An Introduction, Graduate Texts in Mathematics, 217, Springer-Verlag, New York, 2002 | MR | Zbl

[27] E. Hrushovski, “A new strongly minimal set”, Ann. Pure Appl. Logic, 62:2 (1993), 147–166 | DOI | MR | Zbl

[28] A.R. Yeshkeyev, M.T. Omarova, G.E. Zhumabekova, “The $J$-minimal sets in the hereditary theories”, Bulletin of the Karaganda University. Mathematics Series, 2(94) (2019), 92–98 | DOI

[29] J.T. Baldwin, D.W. Kueker, “Algebraically prime models”, Ann. Math. Logic, 20 (1981), 289–330 | DOI | MR | Zbl