{
  "id": "2402.11619",
  "version": "v2",
  "published": "2024-02-18T15:22:31.000Z",
  "updated": "2024-08-10T02:37:07.000Z",
  "title": "Relativized Galois groups of first order theories over a hyperimaginary",
  "authors": [
    "Hyoyoon Lee",
    "Junguk Lee"
  ],
  "comment": "19 pages; many omitted proofs are now provided, exposition is improved, half of the last section is rewritten, citations and typos are corrected",
  "categories": [
    "math.LO"
  ],
  "abstract": "We study relativized Lascar groups, which are formed by relativizing Lascar groups to the solution set of a partial type $\\Sigma$. We introduce the notion of a Lascar tuple for $\\Sigma$ and by considering the space of types over a Lascar tuple for $\\Sigma$, 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 $\\Sigma$. Using this, we find the correspondence between subgroups of the relativized Lascar group and the relativized strong types.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2024-08-10T02:37:07.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "03C99"
    ],
    "keywords": [
      "first order theories",
      "relativized galois groups",
      "hyperimaginary",
      "solution set",
      "partial type"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}