{
  "id": "2112.07518",
  "version": "v2",
  "published": "2021-12-14T16:24:47.000Z",
  "updated": "2022-10-25T08:54:42.000Z",
  "title": "Polyhedral completeness of intermediate logics: the Nerve Criterion",
  "authors": [
    "Sam Adam-Day",
    "Nick Bezhanishvili",
    "David Gabelaia",
    "Vincenzo Marra"
  ],
  "comment": "37 pages, 15 figures",
  "categories": [
    "math.LO",
    "math.CO",
    "math.GT"
  ],
  "abstract": "We investigate a recently-devised polyhedral semantics for intermediate logics, in which formulas are interpreted in n-dimensional polyhedra. An intermediate logic is polyhedrally complete if it is complete with respect to some class of polyhedra. The first main result of this paper is a necessary and sufficient condition for the polyhedral-completeness of a logic. This condition, which we call the Nerve Criterion, is expressed in terms of Alexandrov's notion of the nerve of a poset. It affords a purely combinatorial characterisation of polyhedrally-complete logics. Using the Nerve Criterion we show, easily, that there are continuum many intermediate logics that are not polyhedrally-complete but which have the finite model property. We also provide, at considerable combinatorial labour, a countably infinite class of logics axiomatised by the Jankov-Fine formulas of 'starlike trees' all of which are polyhedrally-complete. The polyhedral completeness theorem for these 'starlike logics' is the second main result of this paper.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2022-10-25T08:54:42.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "03B55",
      "52B05",
      "06A07",
      "03B45",
      "06D20"
    ],
    "keywords": [
      "intermediate logic",
      "nerve criterion",
      "second main result",
      "polyhedral completeness theorem",
      "first main result"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 37,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}