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 Algebra Logika, 2004, Volume 43, Number 4, Pages 459–481 (Mi al83)

$\Sigma$-Definability in Hereditarily Finite Superstructures and Pairs of Models

A. I. Stukachev

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

Abstract: We consider the problem of being $\Sigma$-definable for an uncountable model of a $c$-simple theory in hereditarily finite superstructures over models of another $c$-simple theory. A necessary condition is specified in terms of decidable models and the concept of relative indiscernibility introduced in the paper. A criterion is stated for the uncountable model of a $c$-simple theory to be $\Sigma$-definable in superstructures over dense linear orders, and over infinite models of the empty signature. We prove the existence of a $c$-simple theory (of an infinite signature) every uncountable model of which is not $\Sigma$-definable in superstructures over dense linear orders. Also, a criterion is given for a pair of models to be recursively saturated.

Keywords: $\Sigma$-definability, $c$-simple theory, model, hereditarily finite superstructure, linear order

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English version:
Algebra and Logic, 2004, 43:4, 258–270

Bibliographic databases:

UDC: 510.5

Citation: A. I. Stukachev, “$\Sigma$-Definability in Hereditarily Finite Superstructures and Pairs of Models”, Algebra Logika, 43:4 (2004), 459–481; Algebra and Logic, 43:4 (2004), 258–270

Citation in format AMSBIB
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\by A.~I.~Stukachev
\paper $\Sigma$-Definability in Hereditarily Finite Superstructures and Pairs of Models
\jour Algebra Logika
\yr 2004
\vol 43
\issue 4
\pages 459--481
\mathnet{http://mi.mathnet.ru/al83}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=2105849}
\zmath{https://zbmath.org/?q=an:1063.03020}
\transl
\jour Algebra and Logic
\yr 2004
\vol 43
\issue 4
\pages 258--270
\crossref{https://doi.org/10.1023/B:ALLO.0000035117.75573.2d}
\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-26444473522}

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. Stukachev A., “Presentations of structures in admissible sets”, New Computational Paradigms, Lecture Notes in Computer Science, 3526, 2005, 470–478
2. A. I. Stukachev, “$\Sigma$-definability of uncountable models of $c$-simple theories”, Siberian Math. J., 51:3 (2010), 515–524
3. V. G. Puzarenko, “Countably categorical theories”, Algebra and Logic, 51:3 (2012), 241–258
4. A. S. Morozov, “$\Sigma$-preorderings in ${\mathbb{HF}(\mathbb{R})}$”, Algebra and Logic, 58:5 (2019), 405–416
5. A. I. Stukachev, “Intervalnye rasshireniya poryadkov i temporalnye approksimatsionnye prostranstva”, Sib. matem. zhurn., 62:4 (2021), 894–910
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