Fraïssé limit via forcing
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 aFra\"{i}ss\'{e} class if it satisfies Joint Embedding Property (JEP) and Amalgamation Property (AP). A Fra\"...
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| Main Author: | |
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| Format: | Article |
| Language: | English |
| Published: |
Shahid Bahonar University of Kerman
2024-12-01
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| Series: | Journal of Mahani Mathematical Research |
| Subjects: | |
| Online Access: | https://jmmrc.uk.ac.ir/article_4112_f817a30f4eb6c933680a5926059f0e3e.pdf |
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| Summary: | 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 aFra\"{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 aFra\"{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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| ISSN: | 2251-7952 2645-4505 |