{
  "id": "2011.07399",
  "version": "v1",
  "published": "2020-11-14T22:02:29.000Z",
  "updated": "2020-11-14T22:02:29.000Z",
  "title": "A type of algebraic structure related to sets of intervals",
  "authors": [
    "George M. Bergman"
  ],
  "comment": "11 pages. Copy at http://math.berkeley.edu/~gbergman/papers may be updated more frequently than arXiv copy. This work is far from my field of expertise, and I would welcome comments from those closer to field than I, both on the content and on points of language etc",
  "categories": [
    "math.CO",
    "math.RA"
  ],
  "abstract": "F. Wehrung has asked: Given a family $\\mathcal{C}$ of subsets of a set $\\Omega$, under what conditions will there exist a total ordering on $\\Omega$ under which every member of $\\mathcal{C}$ is convex? <p> Note that if $A$ and $B$ are nondisjoint convex subsets of a totally ordered set, neither of which contains the other, then $A\\cup B$, $A\\cap B$, and $A\\setminus B$ are also convex. So let $\\mathcal{C}$ be an arbitrary set of subsets of a set $\\Omega$, and form its closure $\\mathcal{P}$ under forming, whenever $A$ and $B$ are nondisjoint and neither contains the other, the sets $A\\cup B$, $A\\cap B$, and $A\\setminus B$. We determine the form $\\mathcal{P}$ can take when $\\mathcal{C}$, and hence $\\mathcal{P}$, is finite, and for this case get necessary and sufficient conditions for there to exist an ordering of $\\Omega$ of the desired sort. From this we obtain a condition which works without the finiteness hypothesis. <p> We establish bounds on the cardinality of the subset $\\mathcal{P}$ generated as above by an $n$-element set $\\mathcal{C}$. <p> We note connections with the theory of <i>interval graphs</i> and <i>hypergraphs</i>, which lead to other ways of answering Wehrung's question.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-11-14T22:02:29.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "06A05",
      "08A05",
      "08A55"
    ],
    "keywords": [
      "algebraic structure",
      "nondisjoint convex subsets",
      "sufficient conditions",
      "finiteness hypothesis",
      "element set"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 11,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}