{
  "id": "1811.01754",
  "version": "v1",
  "published": "2018-11-02T10:35:58.000Z",
  "updated": "2018-11-02T10:35:58.000Z",
  "title": "Back and forth between algebraic geometry, algebraic logic, sheaves and forcing",
  "authors": [
    "Trek Sayed Ahmed"
  ],
  "comment": "arXiv admin note: substantial text overlap with arXiv:1304.0612",
  "categories": [
    "math.LO"
  ],
  "abstract": "We study sheaves in the context of a duality theory for lattice structure endowed with extra operations, and in the context of forcing in a topos. Using Sheaf duality theory of Comer for cylindric algebras, we give a representation theorem of of distributive bounded lattices expanded by modalities (functions distributing over joins) as the continuous sections of sheaves. Our representation is defined via a contravariant functor from an algebraic category to a category of sheaves. We show that if our category is a small site (cartesian closed with a stability condition on pullbacks), then we can define a notion of forcing using this category. In particular, we define fuzzy forcing by interpreting the additional Lukasiewicz conjunction $\\otimes$ as induced by a tensor product in the target monodial category of pre-sheaves. We also study topoi as semantics for higher order logic of many sorted theories in connection to set theory, and the quasi-topoi based on $MV$ algebras, for fuzzy logic. We show that the interpretation of a theory $T$, in this case into $Set_{\\Omega}$ where $\\Omega$ is an almost sub-object classifier in a quasi-topos $CAT$ defined from $T$, is completed by defining semantics for $\\otimes$, and this is done similarly to its defining clause in forcing. We give applications to many-valued logics and various modifications of first order logic and multi-modal logic, set in an algebraic framework.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-11-02T10:35:58.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "algebraic logic",
      "algebraic geometry",
      "additional lukasiewicz conjunction",
      "target monodial category",
      "higher order logic"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}