{
  "id": "2004.00989",
  "version": "v1",
  "published": "2020-04-02T13:38:50.000Z",
  "updated": "2020-04-02T13:38:50.000Z",
  "title": "Lattices of Intermediate Theories via Ruitenburg's Theorem",
  "authors": [
    "Gianluca Grilletti",
    "Davide Emilio Quadrellaro"
  ],
  "categories": [
    "math.LO"
  ],
  "abstract": "For every univariate formula $\\chi$ we introduce a lattices of intermediate theories: the lattice of $\\chi$-logics. The key idea to define chi-logics is to interpret atomic propositions as fixpoints of the formula $\\chi^2$, which can be characterised syntactically using Ruitenburg's theorem. We develop an algebraic duality between the lattice of $\\chi$-logics and a special class of varieties of Heyting algebras. This approach allows us to build five distinct lattices corresponding to the possible fixpoints of univariate formulas|among which the lattice of negative variants of intermediate logics. We describe these lattices in more detail.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-04-02T13:38:50.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "ruitenburgs theorem",
      "intermediate theories",
      "interpret atomic propositions",
      "intermediate logics",
      "distinct lattices"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}