{
  "id": "2309.16997",
  "version": "v1",
  "published": "2023-09-29T05:50:44.000Z",
  "updated": "2023-09-29T05:50:44.000Z",
  "title": "Expressive Power of Infinitary Logic and Absolute co-Hopfianity",
  "authors": [
    "Mohsen Asgharzadeh",
    "Mohammad Golshani",
    "Saharon Shelah"
  ],
  "categories": [
    "math.LO",
    "math.AC",
    "math.GR"
  ],
  "abstract": "Paolini and Shelah have constructed absolutely Hopfian torsion-free abelian groups of any given size. In contrast, we show that this is not necessarily the case for absolutely co-Hopfian groups. We apply the infinitary logic to show a dichotomy behavior of co-Hopfian groups, by showing that the first beautiful cardinal is the turning point for this property. Namely, we prove that there are no absolute co-Hopfian abelian groups above the first beautiful cardinal. An extension of this result to the category of modules over a commutative ring is given.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-09-29T05:50:44.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "infinitary logic",
      "absolute co-hopfianity",
      "expressive power",
      "first beautiful cardinal",
      "co-hopfian groups"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}