{
  "id": "1408.3282",
  "version": "v1",
  "published": "2014-06-25T16:58:28.000Z",
  "updated": "2014-06-25T16:58:28.000Z",
  "title": "Atom-canonicity and complete representations for cylindric-like algebras, and omitting types for the clque guarded fragment of first order logic",
  "authors": [
    "Tarek Sayed Ahmed"
  ],
  "comment": "arXiv admin note: text overlap with arXiv:1308.6165, arXiv:1307.1016",
  "categories": [
    "math.LO"
  ],
  "abstract": "Fix a finite ordinal n>2. We show that there exists an atomic, simple and countable representable CA_n, such that its minimal completion is outside SNr_nCA_{n+3}. Hence, for any finite k\\geq 3, the variety SNr_nCA_{n+k} is not atom-canonical, so that the variety of CA_n's having n+k-flat representations is not atom-canonical, too. We show, for finite k\\geq 3, that S_cNr_nCA_{n+k} is not elementary, hence the class of CA_n's having complete n+3-smooth representations is not elementary. We obtain analogous results by replacing flat and smooth, respectively, by (the weaker notion of) square; this give a stronger result in both cases and here we can allow k to be infinite. Our results are proved using rainbow constructions for CA's. We lift the negative result on atom-canonicity to the transfinite. We also show that for any ordinal \\alpha\\geq \\omega, for any finite k\\geq 1, and for any r\\in \\omega, there exists an atomic algebra A_r\\in SNr_\\alphaCA_{\\alpha+k}\\sim SNr_nCA_{\\alpha+k+1}, such that \\Pi_{r/U} A_r\\in RCA_{\\alpha} where U is any non--principal ultrafilter on \\omega. Reaping the harvest of our algebraic results we investigate a plethora of omitting types theorems for variants of first logic including its finite variable fragments and its packed fragment.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-06-25T16:58:28.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "first order logic",
      "clque guarded fragment",
      "omitting types",
      "complete representations",
      "cylindric-like algebras"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1408.3282S"
    }
  }
}