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Sibirskii Matematicheskii Zhurnal, 2009, Volume 50, Number 3, Pages 625–630
(Mi smj1987)
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This article is cited in 1 scientific paper (total in 1 paper)
On expansions and extensions of powerful digraphs
S. V. Sudoplatov Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, Novosibirsk
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 $\exists$-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 $p_1(x),\dots,p_n(x)$ of all nonprincipal 1-types in an LCC theory, if all types $r(x_1,\dots,x_m)$ containing $p_{i_1}(x_1)\cup\cdots\cup p_{i_m}(x_m)$ are dominated by some type $q$ then $q$ is a powerful type.
Keywords:
powerful digraph, powerful type, type-stable theory, Ehrenfeucht theory, locally countably categorical theory.
Received: 22.05.2007
Citation:
S. V. Sudoplatov, “On expansions and extensions of powerful digraphs”, Sibirsk. Mat. Zh., 50:3 (2009), 625–630; Siberian Math. J., 50:3 (2009), 498–502
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https://www.mathnet.ru/eng/smj1987 https://www.mathnet.ru/eng/smj/v50/i3/p625
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Abstract page: | 293 | Full-text PDF : | 68 | References: | 64 | First page: | 4 |
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