Fraïssé limit via forcing

Document Type : Special Issue Dedicated to Prof. Esfandiar Eslami

Author

School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P.O. Box: 19395-5746, Tehran-Iran.

Abstract

Suppose $\mathcal{L}$ is a finite relational language and $\mathcal{K}$ is a class of finite $\mathcal{L}$-structures closed under substructures and isomorphisms. It is called a
Fra\"{i}ss'{e} class if it satisfies Joint Embedding Property (JEP) and Amalgamation Property (AP). A Fra\"{i}ss'{e} limit, denoted $Flim(\mathcal{K})$, of a
Fra\"{i}ss'{e} class $\mathcal{K}$ is the unique\footnote{The existence and uniqueness follows from Fra\"{i}ss'{e}'s theorem, See \cite{hodges}.} countable ultrahomogeneous (every isomorphism of finitely-generated substructures extends to an automorphism of $Flim(\mathcal{K})$) structure into which every member of $\mathcal{K}$ embeds.

Given a Fraïssé class K and an infinite cardinal κ, we define a forcing notion which adds a structure of size κ using elements of K, which extends the Fraïssé construction in the case κ=ω.

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References

[1] Ackerman, Nathanael; Golshani, Mohammad; Mirabi, Mostafa; Cohen Generic Structures with Functions, preprint,
[2] Hodges, Wilfrid Model theory. Encyclopedia of Mathematics and its Applications, 42. Cambridge University Press, Cambridge, 1993. xiv+772 pp. ISBN: 0-521-30442-3
[3] Kostana, Ziemowit; Forcing-theoretic framework for the Frasse theory, PhD thesis.
[4] Kostana, Ziemowit; Cohen-like  rst order structures, Ann. Pure Appl. Logic174(2023), no.1, Paper No. 103172, 17 pp.
Journal of Mahani Mathematical Research

Articles in Press, Accepted Manuscript
Available Online from 18 January 2024

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History

  • Receive Date: 07 November 2023
  • Revise Date: 30 December 2023
  • Accept Date: 18 January 2024

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