{
  "id": "2109.08943",
  "version": "v1",
  "published": "2021-09-18T14:44:43.000Z",
  "updated": "2021-09-18T14:44:43.000Z",
  "title": "Theories admitting congruences over sets and boundedness",
  "authors": [
    "Samuel Braunfeld",
    "Michael C Laskowski"
  ],
  "comment": "6 pages",
  "categories": [
    "math.LO"
  ],
  "abstract": "We consider several ways of decomposing models into parts of bounded size forming a congruence over a base, and show that admitting any such decomposition is equivalent to mutual algebraicity at the level of theories. We also show that a theory $T$ is mutually algebraic if and only if for every $M \\preceq N \\models T$, there is an absolute bound on the number of types realized in $N-M$ over $M$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-09-18T14:44:43.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "03C45"
    ],
    "keywords": [
      "theories admitting congruences",
      "boundedness",
      "mutual algebraicity",
      "absolute bound",
      "decomposition"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 6,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}