{
  "id": "2003.09144",
  "version": "v1",
  "published": "2020-03-20T08:30:58.000Z",
  "updated": "2020-03-20T08:30:58.000Z",
  "title": "Closures of Union-Closed Families",
  "authors": [
    "Dhruv Bhasin"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "Given a Union-Closed family, which is not equal to the power set of its universe, say $[n]$, one can always add a new set $A\\subsetneq[n]$ to it, such that the new family remains Union-Closed. We construct a new family by collecting all such $A$'s and call this family the closure of $\\mathcal F$. We study various properties of this closure. We characterize families whose closure becomes the power set of $[n]$ and give a checking criteria of closure roots of such families, i.e., existence of $\\mathcal H$ such that closure of $\\mathcal H=\\mathcal F$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-03-20T08:30:58.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "union-closed family",
      "power set",
      "closure roots",
      "family remains"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}