Relativized Galois groups of first order theories over a hyperimaginary
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866909283468181504 |
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| author | Lee, Hyoyoon Lee, Junguk |
| author_facet | Lee, Hyoyoon Lee, Junguk |
| contents | We study relativized Lascar groups, which are formed by relativizing Lascar groups to the solution set of a partial type $Σ$. We introduce the notion of a Lascar tuple for $Σ$ and by considering the space of types over a Lascar tuple for $Σ$, the topology for a relativized Lascar group is (re-)defined and some fundamental facts about the Galois groups of first-order theories are generalized to the relativized context. In particular, we prove that any closed subgroup of a relativized Lascar group corresponds to a stabilizer of a bounded hyperimaginary having at least one representative in the solution set of the given partial type $Σ$. Using this, we find the correspondence between subgroups of the relativized Lascar group and the relativized strong types. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_11619 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Relativized Galois groups of first order theories over a hyperimaginary Lee, Hyoyoon Lee, Junguk Logic 03C99 (Primary) We study relativized Lascar groups, which are formed by relativizing Lascar groups to the solution set of a partial type $Σ$. We introduce the notion of a Lascar tuple for $Σ$ and by considering the space of types over a Lascar tuple for $Σ$, the topology for a relativized Lascar group is (re-)defined and some fundamental facts about the Galois groups of first-order theories are generalized to the relativized context. In particular, we prove that any closed subgroup of a relativized Lascar group corresponds to a stabilizer of a bounded hyperimaginary having at least one representative in the solution set of the given partial type $Σ$. Using this, we find the correspondence between subgroups of the relativized Lascar group and the relativized strong types. |
| title | Relativized Galois groups of first order theories over a hyperimaginary |
| topic | Logic 03C99 (Primary) |
| url | https://arxiv.org/abs/2402.11619 |