{
  "id": "math/0211204",
  "version": "v2",
  "published": "2002-11-13T14:39:51.000Z",
  "updated": "2004-04-19T00:55:02.000Z",
  "title": "Representation functions of additive bases for abelian semigroups",
  "authors": [
    "Melvyn B. Nathanson"
  ],
  "comment": "10 pages. Revised version of paper to appear in Int. J. Math. Math. Sci",
  "categories": [
    "math.NT",
    "math.CO"
  ],
  "abstract": "Let X = S \\oplus G, where S is a countable abelian semigroup and G is a countably infinite abelian group such that {2g : g in G} is infinite. Let pi: X \\to G be the projection map defined by pi(s,g) = g for all x =(s,g) in X. Let f:X \\to N_0 cup infty be any map such that the set pi(f^{-1}(0)) is a finite subset of G. Then there exists a set B contained in X such that r_B(x) = f(x) for all x in X, where the representation function r_B(x) counts the number of sets {x',x''} contained in B such that x' \\neq x'' and x'+x''=x. In particular, every function f from the integers Z into N_0 \\cup infty such that f^{-1}(0) is finite is the representation function of an asymptotic basis for Z.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2004-04-19T00:55:02.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11B13",
      "11B34",
      "11B05",
      "05A30"
    ],
    "keywords": [
      "representation function",
      "additive bases",
      "countably infinite abelian group",
      "countable abelian semigroup",
      "projection map"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 10,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2002math.....11204N"
    }
  }
}