{
  "id": "2401.02302",
  "version": "v1",
  "published": "2024-01-04T14:46:15.000Z",
  "updated": "2024-01-04T14:46:15.000Z",
  "title": "A survey on divisibility of ultrafilters",
  "authors": [
    "Boris Sobot"
  ],
  "categories": [
    "math.LO"
  ],
  "abstract": "An extension of the divisibility relation on $\\mathbb{N}$ to the set $\\beta\\mathbb{N}$ of ultrafilters on $\\mathbb{N}$ was defined and investigated in several papers during the last ten years. Here we make a survey of results obtained so far, adding a few simple results connecting the themes of different stages of the research. The main highlights include: separation into the lower part $L$ (with its division into levels) and the upper part; identifying basic ingredients (powers of primes) and fragmentation of each ultrafilter into them; finding the corresponding upward closed sets belonging to an ultrafilter; estimating cardinalities of divisibility-equivalence classes; extending the congruence relation (in two ways) and checking properties of the obtained relations.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-01-04T14:46:15.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11U10",
      "54D35",
      "54D80",
      "03H15"
    ],
    "keywords": [
      "ultrafilter",
      "upward closed sets belonging",
      "divisibility relation",
      "simple results",
      "upper part"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}