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 Algebra Logika, 2006, Volume 45, Number 3, Pages 314–353 (Mi al148)

Complete Theories with Finitely Many Countable Models. II

S. V. Sudoplatov

Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences

Abstract: Previously, we obtained a syntactic characterization for the class of complete theories with finitely many pairwise non-isomorphic countable models [1]. The most essential part of that characterization extends to Ehrenfeucht theories (i.e., those having finitely many (but more than 1) pairwise non-isomorphic countable models). As the basic parameters defining a finite number of countable models, Rudin–Keisler quasiorders are treated as well as distribution functions defining the number of limit models for equivalence classes w.r.t. these quasiorders. Here, we argue to state that all possible parameters given in the characterization theorem in [1] are realizable. Also, we describe Rudin–Keisler quasiorders in arbitrary small theories. The construction of models of Ehrenfeucht theories with which we come up in the paper is based on using powerful digraphs which, along with powerful types in Ehrenfeucht theories, always locally exist in saturated models of these theories.

Keywords: complete theory, Ehrenfeucht theory, number of countable models, Rudin–Keisler quasiorder

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English version:
Algebra and Logic, 2006, 45:3, 180–200

Bibliographic databases:

UDC: 510.67
Revised: 01.03.2006

Citation: S. V. Sudoplatov, “Complete Theories with Finitely Many Countable Models. II”, Algebra Logika, 45:3 (2006), 314–353; Algebra and Logic, 45:3 (2006), 180–200

Citation in format AMSBIB
\Bibitem{Sud06} \by S.~V.~Sudoplatov \paper Complete Theories with Finitely Many Countable Models. II \jour Algebra Logika \yr 2006 \vol 45 \issue 3 \pages 314--353 \mathnet{http://mi.mathnet.ru/al148} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2289092} \zmath{https://zbmath.org/?q=an:1115.03025} \transl \jour Algebra and Logic \yr 2006 \vol 45 \issue 3 \pages 180--200 \crossref{https://doi.org/10.1007/s10469-006-0016-5} \scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-33745591912} 

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This publication is cited in the following articles:
1. S. V. Sudoplatov, “Syntactic approach to constructions of generic models”, Algebra and Logic, 46:2 (2007), 134–146
2. S. V. Sudoplatov, “Small Stable Generic Graphs with Infinite Weight. Digraphs without Furcations”, Siberian Adv. Math., 18:2 (2008), 142–150
3. S. V. Sudoplatov, “On the number of countable models of complete theories with finite Rudin–Keisler preorders”, Siberian Math. J., 48:2 (2007), 334–338
4. S. V. Sudoplatov, “On expansions and extensions of powerful digraphs”, Siberian Math. J., 50:3 (2009), 498–502
5. S. V. Sudoplatov, “Hypergraphs of prime models and distributions of countable models of small theories”, J. Math. Sci., 169:5 (2010), 680–695
6. Gavryushkin A., “On Constructive Models of Theories with Linear Rudin-Keisler Ordering”, J. Logic Comput., 22:4, SI (2012), 793–805
7. I. V. Shulepov, S. V. Sudoplatov, “Algebras of distributions for isolating formulas of a complete theory”, Sib. elektron. matem. izv., 11 (2014), 380–407
8. R. A. Popkov, S. V. Sudoplatov, “Distributions of countable models of theories with continuum many types”, Sib. elektron. matem. izv., 12 (2015), 267–291
9. S. V. Sudoplatov, “Families of language uniform theories and their generating sets”, Izvestiya Irkutskogo gosudarstvennogo universiteta. Seriya Matematika, 17 (2016), 62–76
10. B. Sh. Kulpeshov, S. V. Sudoplatov, “Linearly Ordered Theories which are Nearly Countably Categorical”, Math. Notes, 101:3 (2017), 475–483
11. Sudoplatov S.V. Kiouvrekis Y. Stefaneas P., “Generic Constructions and Generic Limits”, Algebraic Modeling of Topological and Computational Structures and Applications, Springer Proceedings in Mathematics & Statistics, 219, ed. Lambropoulou S. Theodorou D. Stefaneas P. Kauffman L., Springer, 2017, 375–398
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