Abstract
We study the problem of expanding and extending the structure of a stable powerful digraph to the structure of a stable Ehrenfeucht theory. We define the concepts of type unstability and type strict order property. We establish the presence of the type strict order property for every acyclic graph structure with an infinite chain. The simplest form of expansion of a powerful digraph to the structure of an Ehrenfeucht theory is the expansion with a 1-inessential ordered coloring and locally graph ∃-definable many-placed relations, which enable us to mutually realize nonprincipal types; we prove that this expansion is incapable of keeping the structure in the class of stable structures, and moreover, by the type strict order property it generates the first-order definable strict order property. We define the concept of a locally countably categorical theory (LCC theory) and prove that given the list p 1(x), ..., p n (x) of all nonprincipal 1-types in an LCC theory, if all types r(x 1, ..., x m ) containing \( p_{i_1 } \) (x 1) ∪ ... ∪ \( p_{i_m } \)(x m ) are dominated by some type q then q is a powerful type.
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Sudoplatov S. V., “Powerful digraphs,” Siberian Math. J., 48, No. 1, 165–171 (2007).
Sudoplatov S. V., “Complete theories with finitely many countable models. II,” Algebra and Logic, 45, No. 3, 180–200 (2006).
Sudoplatov S. V., “Inessential combinations and colorings of models,” Siberian Math. J., 44, No. 5, 883–890 (2003).
Sudoplatov S. V., “Complete theories with finitely many countable models. I,” Algebra and Logic, 43, No. 1, 110–124 (2004).
Ikeda K., Pillay A., and Tsuboi A., “On theories having three countable models,” Math. Logic Quart., 44, 161–166 (1998).
Tsuboi A., “On theories having a finite number of nonisomorphic countable models,” J. Symbolic Logic, 50, No. 3, 806–808 (1985).
Woodrow R. E., “A note on countable complete theories having three isomorphism types of count,” J. Symbolic Logic, 41, 672–680 (1976).
Woodrow R. E., “Theories with a finite number of countable models,” J. Symbolic Logic, 43, No. 3, 442–455 (1978).
Handbook of Mathematical Logic. Vol. 1: Model Theory [Russian translation], Nauka, Moscow (1982).
Shelah S., Classification Theory and the Number of Non-Isomorphic Models, North-Holland, Amsterdam (1990) (Stud. Logic Found. Math.; 92).
Sudoplatov S. V. and Ovchinnikova E. V., Discrete Mathematics [in Russian], INFRA-M, Moscow; NGTU, Novosibirsk (2007).
Sudoplatov S. V., “On powerful types in small theories,” Siberian Math. J., 31, No. 4, 118–128 (1990).
Sudoplatov S. V., “Syntactic approach to constructions of generic models,” Algebra and Logic, 46, No. 2, 134–146 (2007).
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Original Russian Text Copyright © 2009 Sudoplatov S. V.
The author was supported by the Russian Foundation for Basic Research (Grants 05-01-00411; 09-01-00336) and the State Maintenance Program for the Leading Scientific Schools of the Russian Federation (Grants NSh-4787.2006.1; NSh-344.2008.1).
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Novosibirsk. Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 50, No. 3, pp. 625–630, May–June, 2009.
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Sudoplatov, S.V. On expansions and extensions of powerful digraphs. Sib Math J 50, 498–502 (2009). https://doi.org/10.1007/s11202-009-0056-x
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DOI: https://doi.org/10.1007/s11202-009-0056-x