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Algebra Logika, 2018 Volume 57, Number 5, Pages 567–586 (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

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.

UDC: 510.67

Received: 06.01.2017

DOI: 10.33048/alglog.2018.57.505


 English version:
Algebra and Logic, 2018, 57:5, 368–380

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