{
  "id": "1609.09149",
  "version": "v1",
  "published": "2016-09-28T22:56:43.000Z",
  "updated": "2016-09-28T22:56:43.000Z",
  "title": "Which semifields are exact?",
  "authors": [
    "Yaroslav Shitov"
  ],
  "comment": "10 pages",
  "categories": [
    "math.CO",
    "math.RA"
  ],
  "abstract": "Every (left) linear function on a subspace of a finite-dimensional vector space over a (skew) field can be extended to a (left) linear function on the whole space. This paper explores the extent to what this basic fact of linear algebra is applicable to more general structures. Semifields with a similar property imposed on linear functions are called (left) exact, and we present a complete description of such semifields. Namely, we show that a semifield $S$ is left exact if and only if $S$ is either a skew field or an idempotent semiring. In particular, our result is new even for the tropical semiring and gives a solution to the problem posed by Wilding. Also, we point out several problems that require further investigation.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-09-28T22:56:43.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "16Y60",
      "15A80"
    ],
    "keywords": [
      "linear function",
      "finite-dimensional vector space",
      "linear algebra",
      "left exact",
      "complete description"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 10,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}