Algebra and Logic

, Volume 57, Issue 5, pp 368–380 | Cite as

Forcing Formulas in Fraïssé Structures and Classes

  • A. T. NurtazinEmail author

We come up with a semantic method of forcing formulas by finite structures in an arbitrary fixed Fraïssé class Open image in new window . Both known and some new necessary and sufficient conditions are derived under which a given structure Open image in new window will be a forcing structure. A formula φ is forced on \( \overline{a} \) in an infinite structure Open image in new window ╟φ\( \left(\overline{a}\right) \) if it is forced in Open image in new window by some finite substructure of Open image in new window . It is proved that every ∃∀∃-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.


forcing method Fraïssé class forcing structure forcing type existentially closed structure existentially closed companion 


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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institute of Information and Computational Technologies, Ministry of Education and Science RKAlma-AtaKazakhstan

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