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 Algebra Logika, 2016, Volume 55, Number 3, Pages 366–379 (Mi al746)

A sufficient condition for nonpresentability of structures in hereditarily finite superstructures

A. S. Morozovab

a Sobolev Institute of Mathematics, pr. Akad. Koptyuga 4, Novosibirsk, 630090 Russia
b Novosibirsk State University, ul. Pirogova 2, Novosibirsk, 630090 Russia

Abstract: We introduce a class of existentially Steinitz structures containing, in particular, the fields of real and complex numbers. A general result is proved which implies that if $\mathfrak M$ is an existentially Steinitz structure then the following structures cannot be embedded in any structure $\Sigma$-presentable with trivial equivalence over $\mathbb{HF}(\mathfrak M)$: the Boolean algebra of all subsets of $\omega$, its factor modulo the ideal consisting of finite sets, the group of all permutations on $\omega$, its factor modulo the subgroup of all finitary permutations, the semigroup of all mappings from $\omega$ to $\omega$, the lattice of all open sets of real numbers, the lattice of all closed sets of real numbers, the group of all permutations of $\mathbb R$ $\Sigma$-definable with parameters over $\mathbb{HF(R)}$, and the semigroup of such mappings from $\mathbb R$ to $\mathbb R$.

Keywords: existentially Steinitz structure, hereditarily finite superstructure, $\Sigma$-presentability.

 Funding Agency Grant Number Ministry of Education and Science of the Russian Federation ÍØ-860.2014.1 Supported by the Grants Council (under RF President) for State Aid of Leading Scientific Schools, grant NSh-860.2014.1.

DOI: https://doi.org/10.17377/alglog.2016.55.305

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English version:
Algebra and Logic, 2016, 55:3, 242–251

Bibliographic databases:

UDC: 510.65
Revised: 09.10.2015

Citation: A. S. Morozov, “A sufficient condition for nonpresentability of structures in hereditarily finite superstructures”, Algebra Logika, 55:3 (2016), 366–379; Algebra and Logic, 55:3 (2016), 242–251

Citation in format AMSBIB
\Bibitem{Mor16} \by A.~S.~Morozov \paper A sufficient condition for nonpresentability of structures in hereditarily finite superstructures \jour Algebra Logika \yr 2016 \vol 55 \issue 3 \pages 366--379 \mathnet{http://mi.mathnet.ru/al746} \crossref{https://doi.org/10.17377/alglog.2016.55.305} \transl \jour Algebra and Logic \yr 2016 \vol 55 \issue 3 \pages 242--251 \crossref{https://doi.org/10.1007/s10469-016-9392-7} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000385155300005} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84989172962} 

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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, “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, eds. A. Day, M. Fellows, N. Greenberg, B. Khoussainov, A. Melnikov, F. Rosamond, Springler, 2017, 354–365
2. A. S. Morozov, “Nonpresentability of some structures of analysis in hereditarily finite superstructures”, Algebra and Logic, 56:6 (2018), 458–472
3. A. S. Morozov, “O $\Sigma$-predporyadkakh v ${\mathbb{HF}(\mathbb{R})}$”, Algebra i logika, 58:5 (2019), 609–626
4. R. M. Korotkova, O. V. Kudinov, A. S. Morozov, “O vzaimnoi opredelimosti operatsii nad polyami”, Sib. matem. zhurn., 60:6 (2019), 1324–1334
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