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| Format: | Preprint |
| Veröffentlicht: |
2025
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| Online-Zugang: | https://arxiv.org/abs/2506.23586 |
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| _version_ | 1866908428119572480 |
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| author | Paolini, Gianluca Pisciotta, Federico |
| author_facet | Paolini, Gianluca Pisciotta, Federico |
| contents | Generalizing the $ω$-categorical context, we introduce a notion, which we call the Lascar Property, that allows for a fine analysis of the topological isomorphisms between automorphism groups of countable structures satisfying this property. In particular, under the assumption of the Lascar Property, we exhibit a definable Galois correspondence between pointwise stabilizers of finitely generated Galois algebraically closed subsets of $M$ and finitely generated Galois algebraically closed subsets of $M$. We use this to characterize the group of automorphisms of $\mathrm{Aut}(M)$, for $M$ the countable saturated model of $\mathrm{ACF}_0$, $\mathrm{DCF}_0$, or the theory of infinite $\mathrm{K}$-vector spaces, generalizing results of Evans $\&$ Lascar, and Konnerth, while at the same time subsuming the analysis from [11] for $ω$-categorical structures with weak elimination of imaginaries. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_23586 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Galois correspondence for automorphism groups of structures with the Lascar Property Paolini, Gianluca Pisciotta, Federico Logic 20B27, 22A05, 20F28, 03C15 Generalizing the $ω$-categorical context, we introduce a notion, which we call the Lascar Property, that allows for a fine analysis of the topological isomorphisms between automorphism groups of countable structures satisfying this property. In particular, under the assumption of the Lascar Property, we exhibit a definable Galois correspondence between pointwise stabilizers of finitely generated Galois algebraically closed subsets of $M$ and finitely generated Galois algebraically closed subsets of $M$. We use this to characterize the group of automorphisms of $\mathrm{Aut}(M)$, for $M$ the countable saturated model of $\mathrm{ACF}_0$, $\mathrm{DCF}_0$, or the theory of infinite $\mathrm{K}$-vector spaces, generalizing results of Evans $\&$ Lascar, and Konnerth, while at the same time subsuming the analysis from [11] for $ω$-categorical structures with weak elimination of imaginaries. |
| title | A Galois correspondence for automorphism groups of structures with the Lascar Property |
| topic | Logic 20B27, 22A05, 20F28, 03C15 |
| url | https://arxiv.org/abs/2506.23586 |