{
  "id": "2302.02264",
  "version": "v1",
  "published": "2023-02-04T23:57:39.000Z",
  "updated": "2023-02-04T23:57:39.000Z",
  "title": "Some semilattices of definable sets in continuous logic",
  "authors": [
    "James Hanson"
  ],
  "comment": "22 pages",
  "categories": [
    "math.LO"
  ],
  "abstract": "In continuous first-order logic, the union of definable sets is definable but generally the intersection is not. This means that in any continuous theory, the collection of $\\varnothing$-definable sets in one variable forms a join-semilattice under inclusion that may fail to be a lattice. We investigate the question of which semilattices arise as the collection of definable sets in a continuous theory. We show that for any non-trivial finite semilattice $L$ (or, equivalently, any finite lattice $L$), there is a superstable theory $T$ whose semilattice of definable sets is $L$. We then extend this construction to some infinite semilattices. In particular, we show that the following semilattices arise in continuous theories: $\\alpha+1$ and $(\\alpha+1)^\\ast$ for any ordinal $\\alpha$, a semilattice containing an exact pair above $\\omega$, and the lattice of filters in $L$ for any countable meet-semilattice $L$. By previous work of the author, this establishes that these semilattices arise in stable theories. The first two are done in languages of cardinality $\\aleph_0 + |\\alpha|$, and the latter two are done in countable languages.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-02-04T23:57:39.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "03C66"
    ],
    "keywords": [
      "definable sets",
      "continuous logic",
      "semilattices arise",
      "continuous theory",
      "non-trivial finite semilattice"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 22,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}