|
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.
Received: 06.01.2017
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
Linking options:
https://www.mathnet.ru/eng/al867 https://www.mathnet.ru/eng/al/v57/i5/p567
|
Statistics & downloads: |
Abstract page: | 245 | Full-text PDF : | 35 | References: | 41 | First page: | 7 |
|