{
  "id": "2407.10183",
  "version": "v1",
  "published": "2024-07-14T12:59:44.000Z",
  "updated": "2024-07-14T12:59:44.000Z",
  "title": "Boundedly finite-to-one functions",
  "authors": [
    "Xiao Hu",
    "Guozhen Shen"
  ],
  "comment": "8 pages",
  "categories": [
    "math.LO",
    "math.CO"
  ],
  "abstract": "A function is boundedly finite-to-one if there is a natural number $k$ such that each point has at most $k$ inverse images. In this paper, we prove in $\\mathsf{ZF}$ (without the axiom of choice) several results concerning this notion, among which are the following: (1) For each infinite set $A$ and natural number $n$, there is no boundedly finite-to-one function from $\\mathcal{S}(A)$ to $\\mathcal{S}_{\\leq n}(A)$, where $\\mathcal{S}(A)$ is the set of all permutations of $A$ and $\\mathcal{S}_{\\leq n}(A)$ is the set of all permutations of $A$ moving at most $n$ points. (2) For each infinite set $A$, there is no boundedly finite-to-one function from $\\mathcal{B}(A)$ to $\\mathrm{fin}(A)$, where $\\mathcal{B}(A)$ is the set of all partitions of $A$ whose blocks are finite and $\\mathrm{fin}(A)$ is the set of all finite subsets of $A$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-07-14T12:59:44.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "03E10",
      "03E25"
    ],
    "keywords": [
      "boundedly finite-to-one function",
      "natural number",
      "infinite set",
      "finite subsets",
      "inverse images"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 8,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}