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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2604.20501 |
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| _version_ | 1866910157409091584 |
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| author | Sullivan, Rob Winkel, Jeroen |
| author_facet | Sullivan, Rob Winkel, Jeroen |
| contents | Let $M$ be a Fraïssé structure (a countably infinite ultrahomogeneous structure). We refer to the class of structures embeddable in $M$ as the $ω$-age of $M$. We consider the following two properties of $M$: we say that $M$ has a universal automorphism group if, for each $A$ in the $ω$-age of $M$, there is an embedding $\textrm{Aut}(A) \to \textrm{Aut}(M)$, and we say that $M$ has group-extensible $ω$-age if, for each $A$ in the $ω$-age of $M$, there is an embedding $A \to M$ such that each automorphism of the image extends to an automorphism of $M$ and the extension map preserves group composition. It is immediate that if $M$ has group-extensible $ω$-age, then $M$ has a universal automorphism group. We give an example of a Fraïssé structure with a universal automorphism group whose $ω$-age is not group-extensible, showing that the above two properties are not equivalent. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_20501 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | An unusual example of a universal automorphism group Sullivan, Rob Winkel, Jeroen Logic 03C15, 20B27, 03C50, 18A22 Let $M$ be a Fraïssé structure (a countably infinite ultrahomogeneous structure). We refer to the class of structures embeddable in $M$ as the $ω$-age of $M$. We consider the following two properties of $M$: we say that $M$ has a universal automorphism group if, for each $A$ in the $ω$-age of $M$, there is an embedding $\textrm{Aut}(A) \to \textrm{Aut}(M)$, and we say that $M$ has group-extensible $ω$-age if, for each $A$ in the $ω$-age of $M$, there is an embedding $A \to M$ such that each automorphism of the image extends to an automorphism of $M$ and the extension map preserves group composition. It is immediate that if $M$ has group-extensible $ω$-age, then $M$ has a universal automorphism group. We give an example of a Fraïssé structure with a universal automorphism group whose $ω$-age is not group-extensible, showing that the above two properties are not equivalent. |
| title | An unusual example of a universal automorphism group |
| topic | Logic 03C15, 20B27, 03C50, 18A22 |
| url | https://arxiv.org/abs/2604.20501 |