{
  "id": "1209.3248",
  "version": "v1",
  "published": "2012-09-14T16:18:20.000Z",
  "updated": "2012-09-14T16:18:20.000Z",
  "title": "The Euler characteristic of a polyhedron as a valuation on its coordinate vector lattice",
  "authors": [
    "Andrea Pedrini"
  ],
  "comment": "13 pages",
  "categories": [
    "math.MG",
    "math.GT"
  ],
  "abstract": "A celebrated theorem of Hadwiger states that the Euler-Poincar\\'e characteristic is the the unique invariant and continuous valuation on the distributive lattice of compact polyhedra in R^n that assigns value one to each convex non-empty such polyhedron. This paper provides an analogue of Hadwiger's result for finitely presented unital vector lattices (i.e. real vector spaces with a compatible lattice order, also known as Riesz spaces). The vector lattice of continuous and piecewise (affine) linear real-valued functions on a compact polyhedron, with operations defined pointwise from the vector lattice R, is a finitely presented unital vector lattice; and it is a non-trivial fact that all such vector lattices arise in this manner, to within an isomorphism. Each function in such a vector lattice can be written as a linear combination of a subset of distinguished elements that we call vl-Schauder hats. We prove here that the functional that assigns to each non-negative piecewise linear function on the polyhedron the Euler-Poincar\\'e characteristic of its support is the unique vl-valuation (a special class of valuations on vector lattices) that assigns one to each vl-Schauder hat of the vector lattice.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2012-09-14T16:18:20.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "06F20",
      "52B45",
      "46A40"
    ],
    "keywords": [
      "coordinate vector lattice",
      "euler characteristic",
      "unital vector lattice",
      "vl-schauder hat",
      "euler-poincare characteristic"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1209.3248P"
    }
  }
}