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Sibirskie Èlektronnye Matematicheskie Izvestiya [Siberian Electronic Mathematical Reports], 2012, Volume 9, Pages 161–184
(Mi semr346)
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This article is cited in 11 scientific papers (total in 12 papers)
Mathematical logic, algebra and number theory
Conditions for non-symmetric relations of semi-isolation
B. S. Baizhanova, S. V. Sudoplatovb, V. V. Verbovskiyc a Institute of Mathematics, Informatics and Mechanics, ul. Pushkina, 125, 050010, Almaty, Kazakhstan
b Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, pr. Koptyuga, 4,
630090, Novosibirsk, Russia
c Institute for Problems of Informatics and Control Sciences, ul. Pushkina, 125,
050010, Almaty, Kazakhstan
Abstract:
We consider necessary and sufficient conditions for non-symmetric relations of semi-isolation in terms of colorings for neighborhoods of types, quasi-neighborhoods, and the existence of limit models. We show that, for any type $p$ in a small theory, its non-symmetry of isolation is equivalent to the non-symmetry of semi-isolation (where a realization $\bar a$ of $p$ isolates a realization $\bar b$ of $p$ and $\bar b$ does not semi-isolates $\bar a$) and is equivalent to the existence of a limit model over $p$. We generalize the Tsuboi theorem on the absence of Ehrenfeucht unions of pseudo-superstable theories and the Kim theorem on the absence of Ehrenfeucht supersimple theories for unions of pseudo-supersimple theories. We also present a survey of results related to non-symmetric semi-isolation.
Keywords:
relation of semi-isolation, $(p,q)$-preserving formula, Ehrenfeucht theory, powerful type, quasi-neighborhood, coloring of a structure, strict order property, limit model.
Received September 26, 2011, published February 21, 2012
Citation:
B. S. Baizhanov, S. V. Sudoplatov, V. V. Verbovskiy, “Conditions for non-symmetric relations of semi-isolation”, Sib. Èlektron. Mat. Izv., 9 (2012), 161–184
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https://www.mathnet.ru/eng/semr346 https://www.mathnet.ru/eng/semr/v9/p161
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