Omega-categorical limits of betweenness relations and $D$-sets
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| Format: | Preprint |
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2024
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| _version_ | 1866909342291197952 |
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| author | Almazaydeh, Asma Ibrahim Braunfeld, Samuel Macpherson, Dugald |
| author_facet | Almazaydeh, Asma Ibrahim Braunfeld, Samuel Macpherson, Dugald |
| contents | We explore two constructions of oligomorphic Jordan permutation groups preserving a `limit of betweenness relations' and a `limit of $D$-relations', from \cite{bhattmacph2006jordan} and \cite{almazaydeh2021jordan} respectively. Several issues left open in \cite{almazaydeh2021jordan} are resolved. In particular it is shown that the `limit of $D$-relations' is not homogeneous in the given language, but is `homogenizable', that is, there is a homogeneous structure over a finite relational language with the same universe and the same automorphism group. The structure is NIP, but not monadically NIP, its age is not well-quasi-ordered under embeddability, and the growth rate of the sequence enumerating orbits on $k$-sets grows faster than exponentially. The automorphism group is maximal-closed in the symmetric group. Similar results are shown for the construction in \cite{bhattmacph2006jordan}. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_05832 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Omega-categorical limits of betweenness relations and $D$-sets Almazaydeh, Asma Ibrahim Braunfeld, Samuel Macpherson, Dugald Logic Combinatorics Group Theory We explore two constructions of oligomorphic Jordan permutation groups preserving a `limit of betweenness relations' and a `limit of $D$-relations', from \cite{bhattmacph2006jordan} and \cite{almazaydeh2021jordan} respectively. Several issues left open in \cite{almazaydeh2021jordan} are resolved. In particular it is shown that the `limit of $D$-relations' is not homogeneous in the given language, but is `homogenizable', that is, there is a homogeneous structure over a finite relational language with the same universe and the same automorphism group. The structure is NIP, but not monadically NIP, its age is not well-quasi-ordered under embeddability, and the growth rate of the sequence enumerating orbits on $k$-sets grows faster than exponentially. The automorphism group is maximal-closed in the symmetric group. Similar results are shown for the construction in \cite{bhattmacph2006jordan}. |
| title | Omega-categorical limits of betweenness relations and $D$-sets |
| topic | Logic Combinatorics Group Theory |
| url | https://arxiv.org/abs/2410.05832 |