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| Main Authors: | , , |
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| Format: | Preprint |
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2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2508.06370 |
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| _version_ | 1866909738341498880 |
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| author | Kwiatkowska, Aleksandra Sullivan, Rob Winkel, Jeroen |
| author_facet | Kwiatkowska, Aleksandra Sullivan, Rob Winkel, Jeroen |
| contents | Let $M$ be a Fraïssé structure (a countably infinite ultrahomogeneous structure). We call an embedding $f : A \to M$ extensive if each automorphism of its image extends to an automorphism of $M$, where the extension map respects composition, and we say that $M$ has extensible $ω$-age if each substructure admits an extensive embedding into $M$.
We investigate the relationship between the following two properties: the presence of a stationary weak independence relation (SWIR) on $M$, and extensibility of the $ω$-age of $M$. We show that linearly ordered Fraïssé structures with a SWIR have extensible $ω$-age, but also we give examples of Fraïssé structures where only one of the two properties holds. Finally, we consider whether a wide range of examples of Fraïssé structures have extensible $ω$-age or a finite SWIR expansion, including all countably infinite ultrahomogeneous oriented graphs (with one exception). |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2508_06370 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Extensive embeddings into Fraïssé structures and stationary weak independence relations Kwiatkowska, Aleksandra Sullivan, Rob Winkel, Jeroen Logic Combinatorics 03C15, 20B27, 03C50, 18A22 Let $M$ be a Fraïssé structure (a countably infinite ultrahomogeneous structure). We call an embedding $f : A \to M$ extensive if each automorphism of its image extends to an automorphism of $M$, where the extension map respects composition, and we say that $M$ has extensible $ω$-age if each substructure admits an extensive embedding into $M$. We investigate the relationship between the following two properties: the presence of a stationary weak independence relation (SWIR) on $M$, and extensibility of the $ω$-age of $M$. We show that linearly ordered Fraïssé structures with a SWIR have extensible $ω$-age, but also we give examples of Fraïssé structures where only one of the two properties holds. Finally, we consider whether a wide range of examples of Fraïssé structures have extensible $ω$-age or a finite SWIR expansion, including all countably infinite ultrahomogeneous oriented graphs (with one exception). |
| title | Extensive embeddings into Fraïssé structures and stationary weak independence relations |
| topic | Logic Combinatorics 03C15, 20B27, 03C50, 18A22 |
| url | https://arxiv.org/abs/2508.06370 |