{
  "id": "2008.02722",
  "version": "v1",
  "published": "2020-08-06T15:47:18.000Z",
  "updated": "2020-08-06T15:47:18.000Z",
  "title": "Congruence of ultrafilters",
  "authors": [
    "Boris Šobot"
  ],
  "comment": "1 figure",
  "categories": [
    "math.LO"
  ],
  "abstract": "We continue the research of the relation $\\hspace{1mm}\\widetilde{\\mid}\\hspace{1mm}$ on the set $\\beta {\\mathbb{N}}$ of ultrafilters on ${\\mathbb{N}}$, defined as an extension of the divisibility relation. It is a quasiorder, so we see it as an order on the set of $=_\\sim$-equivalence classes, where ${\\cal F}=_\\sim{\\cal G}$ means that ${\\cal F}$ and ${\\cal G}$ are mutually $\\hspace{1mm}\\widetilde{\\mid}\\hspace{1mm}$-divisible. Here we introduce a new tool: a relation of congruence modulo an ultrafilter. We first recall the congruence of ultrafilters modulo an integer and show that $=_\\sim$-equivalent ultrafilters do not necessarily have the same residue modulo $m\\in {\\mathbb{N}}$. Then we generalize this relation to congruence modulo an ultrafilter in a natural way. After that, using iterated nonstandard extensions, we introduce a stronger relation, which has nicer properties with respect to addition and multiplication of ultrafilters. Finally, we also introduce a strengthening of $\\hspace{1mm}\\widetilde{\\mid}\\hspace{1mm}$ and show that it also behaves well in relation to the congruence relation.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-08-06T15:47:18.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "54D35",
      "54D80",
      "11A07",
      "11U10",
      "03H15"
    ],
    "keywords": [
      "congruence modulo",
      "stronger relation",
      "divisibility relation",
      "iterated nonstandard extensions",
      "natural way"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}