{
  "id": "1910.01094",
  "version": "v1",
  "published": "2019-10-02T17:18:58.000Z",
  "updated": "2019-10-02T17:18:58.000Z",
  "title": "More number theory in $βN$",
  "authors": [
    "Boris Šobot"
  ],
  "categories": [
    "math.LO"
  ],
  "abstract": "We continue the research of an extension $\\widetilde{\\mid}$ of the divisibility relation to the Stone-\\v Cech compactification $\\beta N$. First we prove that ultrafilters we call prime actually possess the algebraic property of primality. Several questions concerning the connection between divisibilities in $\\beta N$ and nonstandard extensions of $N$ are answered, providing a few more equivalent conditions for divisibility in $\\beta N$. Results on uncountable chains in $(\\beta N,\\widetilde{\\mid})$ are proved and used in a construction of a well-ordered chain of maximal cardinality. Finally, we consider ultrafilters without divisors in $N$ and among them find the maximal class.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-10-02T17:18:58.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11U10",
      "03H15",
      "54D35",
      "54D80"
    ],
    "keywords": [
      "number theory",
      "divisibility relation",
      "cech compactification",
      "ultrafilters",
      "maximal class"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}