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Algebra i logika, 2018, Volume 57, Number 5, Pages 567–586
DOI: https://doi.org/10.33048/alglog.2018.57.505
(Mi al867)
 

Forcing formulas in Fraïssé structures and classes

A. T. Nurtazin

Institute of Information and Computational Technologies, Ministry of Education and Science RK, ul. Pushkina 125, Alma-Ata, 050010 Kazakhstan
Full-text PDF (205 kB) (1)
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Abstract: We come up with a semantic method of forcing formulas by finite structures in an arbitrary fixed Fraïssé class $\mathscr F$. Both known and some new necessary and sufficient conditions are derived under which a given structure $\mathscr M$ will be a forcing structure. A formula $\varphi$ is forced at $\bar a$ in an infinite structure $\mathscr M\Vdash\varphi(\bar a)$ if it is forced in $\mathscr F(\mathscr M)$ by some finite substructure of $\mathscr M$. It is proved that every $\exists\forall\exists$-sentence true in a forcing structure is also true in any existentially closed companion of the structure.
The new concept of a forcing type plays an important role in studying forcing models. It is proved that an arbitrary structure will be a forcing structure iff all existential types realized in the structure are forcing types. It turns out that an existentially closed structure which is simple over a tuple realizing a forcing type will itself be a forcing structure. Moreover, every forcing type is realized in an existentially closed structure that is a model of a complete theory of its forcing companion.
Keywords: forcing method, Fraïssé class, forcing structure, forcing type, existentially closed structure, existentially closed companion.
Funding agency Grant number
Ministry of Education and Science of the Republic of Kazakhstan 0174/ГФ4
Supported by KN MON RK, project No. 0174/GF4.
Received: 06.01.2017
English version:
Algebra and Logic, 2018, Volume 57, Issue 5, Pages 368–380
DOI: https://doi.org/10.1007/s10469-018-9509-2
Bibliographic databases:
Document Type: Article
UDC: 510.67
Language: Russian
Citation: A. T. Nurtazin, “Forcing formulas in Fraïssé structures and classes”, Algebra Logika, 57:5 (2018), 567–586; Algebra and Logic, 57:5 (2018), 368–380
Citation in format AMSBIB
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\by A.~T.~Nurtazin
\paper Forcing formulas in Fra\"\i ss\'e structures and classes
\jour Algebra Logika
\yr 2018
\vol 57
\issue 5
\pages 567--586
\mathnet{http://mi.mathnet.ru/al867}
\crossref{https://doi.org/10.33048/alglog.2018.57.505}
\transl
\jour Algebra and Logic
\yr 2018
\vol 57
\issue 5
\pages 368--380
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  • https://www.mathnet.ru/eng/al/v57/i5/p567
  • This publication is cited in the following 1 articles:
    Citing articles in Google Scholar: Russian citations, English citations
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    Алгебра и логика Algebra and Logic
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    Full-text PDF :35
    References:41
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