{
  "id": "1706.03276",
  "version": "v1",
  "published": "2017-06-10T20:14:59.000Z",
  "updated": "2017-06-10T20:14:59.000Z",
  "title": "Interval orders, semiorders and ordered groups",
  "authors": [
    "Maurice Pouzet",
    "Imed Zaguia"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "We prove that the order of an ordered group is an interval order if and only if it is a semiorder. Next, we prove that every semiorder is isomorphic to a collection $\\mathcal J$ of intervals of some totally ordered abelian group, these intervals being of the form $[x, x+ \\alpha[$ for some positive $\\alpha$. We describe ordered groups such that the ordering is a semiorder and we introduce threshold groups generalizing totally ordered groups. We show that the free group on finitely many generators and the Thompson group $\\mathbb F$ can be equipped with a semiorder which is not a weak order. On an other hand, a group introduced by Clifford cannot.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-06-10T20:14:59.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "interval order",
      "free group",
      "totally ordered abelian group",
      "weak order",
      "groups generalizing totally ordered groups"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}