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 Algebra Logika, 2008, Volume 47, Number 3, Pages 335–363 (Mi al362)

$\Sigma$-Definability of countable structures over real numbers, complex numbers, and quaternions

A. S. Morozova, M. V. Korovinab

a Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences
b A. P. Ershov Institute of Informatics Systems Sib. Br. RAS

Abstract: We study $\Sigma$-definability of countable models over hereditarily finite ($\mathbb{HF}$-) superstructures over the field $\mathbb R$ of reals, the field $\mathbb C$ of complex numbers, and over the skew field $\mathbb H$ of quaternions. In particular, it is shown that each at most countable structure of a finite signature, which is $\Sigma$-definable over $\mathbb{HF}(\mathbb R)$ with at most countable equivalence classes and without parameters, has a computable isomorphic copy. Moreover, if we lift the requirement on the cardinalities of the classes in a definition then such a model can have an arbitrary hyperarithmetical complexity, but it will be hyperarithmetical in any case. Also it is proved that any countable structure $\Sigma$-definable over $\mathbb{HF}(\mathbb C)$, possibly with parameters, has a computable isomorphic copy and that being $\Sigma$-definable over $\mathbb{HF}(\mathbb H)$ is equivalent to being $\Sigma$-definable over $\mathbb{HF}(\mathbb R)$.

Keywords: countable model, computable model, $\Sigma$-definability.

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English version:
Algebra and Logic, 2008, 47:3, 193–209

Bibliographic databases:

UDC: 510.67+510.5
Revised: 14.02.2008

Citation: A. S. Morozov, M. V. Korovina, “$\Sigma$-Definability of countable structures over real numbers, complex numbers, and quaternions”, Algebra Logika, 47:3 (2008), 335–363; Algebra and Logic, 47:3 (2008), 193–209

Citation in format AMSBIB
\Bibitem{MorKor08} \by A.~S.~Morozov, M.~V.~Korovina \paper $\Sigma$-Definability of countable structures over real numbers, complex numbers, and quaternions \jour Algebra Logika \yr 2008 \vol 47 \issue 3 \pages 335--363 \mathnet{http://mi.mathnet.ru/al362} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2450887} \zmath{https://zbmath.org/?q=an:1164.03332} \transl \jour Algebra and Logic \yr 2008 \vol 47 \issue 3 \pages 193--209 \crossref{https://doi.org/10.1007/s10469-008-9009-x} \scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-49249083112} 

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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. A. S. Morozov, “Nonpresentability of the semigroup $\omega^\omega$ over $\mathbb{HF(R)}$”, Siberian Math. J., 55:1 (2014), 125–131
2. S. A. Aleksandrova, “The uniformization problem for $\Sigma$-predicates in a hereditarily finite list superstructure over the real exponential field”, Algebra and Logic, 53:1 (2014), 1–8
3. A. S. Morozov, “$\Sigma$-presentations of the ordering on the reals”, Algebra and Logic, 53:3 (2014), 217–237
4. A. S. Morozov, “A sufficient condition for nonpresentability of structures in hereditarily finite superstructures”, Algebra and Logic, 55:3 (2016), 242–251
5. Dino Rossegger, “On functors enumerating structures”, Sib. elektron. matem. izv., 14 (2017), 690–702
6. Morozov A.S., “Computable Model Theory Over the Reals”, Computability and Complexity: Essays Dedicated to Rodney G. Downey on the Occasion of His 60Th Birthday, Lecture Notes in Computer Science, 10010, ed. Day A. Fellows M. Greenberg N. Khoussainov B. Melnikov A. Rosamond F., Springer International Publishing Ag, 2017, 354–365
7. A. S. Morozov, “Nonpresentability of some structures of analysis in hereditarily finite superstructures”, Algebra and Logic, 56:6 (2018), 458–472
8. S. A. Aleksandrova, “O $\Sigma$-opredelimosti nasledstvenno konechnoi i spisochnoi nadstroek”, Sib. zhurn. chist. i prikl. matem., 18:1 (2018), 3–10
9. A. S. Morozov, “$\Sigma$-preorderings in ${\mathbb{HF}(\mathbb{R})}$”, Algebra and Logic, 58:5 (2019), 405–416
10. R. M. Korotkova, O. V. Kudinov, A. S. Morozov, “On mutual definability of operations on fields”, Siberian Math. J., 60:6 (2019), 1032–1039
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