{
  "id": "2307.16017",
  "version": "v1",
  "published": "2023-07-29T16:09:30.000Z",
  "updated": "2023-07-29T16:09:30.000Z",
  "title": "Yet another ideal version of the bounding number",
  "authors": [
    "Rafał Filipów",
    "Adam Kwela"
  ],
  "journal": "J. Symb. Log. 87 (2022), no. 3, 1065-1092",
  "doi": "10.1017/jsl.2021.69",
  "categories": [
    "math.LO"
  ],
  "abstract": "Let $\\mathcal{I}$ be an ideal on $\\omega$. For $f,g\\in\\omega^\\omega$ we write $f \\leq_{\\mathcal{I}} g$ if $f(n) \\leq g(n)$ for all $n\\in\\omega\\setminus A$ with some $A\\in\\mathcal{I}$. Moreover, we denote $\\mathcal{D}_{\\mathcal{I}}=\\{f\\in\\omega^\\omega: f^{-1}[\\{n\\}]\\in\\mathcal{I} \\text{ for every $n\\in \\omega$}\\}$ (in particular, $\\mathcal{D}_{Fin}$ denotes the family of all finite-to-one functions). We examine cardinal numbers $\\mathfrak{b}(\\geq_{\\mathcal{I}}\\cap (\\mathcal{D}_{\\mathcal{I}} \\times \\mathcal{D}_{\\mathcal{I}}))$ and $\\mathfrak{b}(\\geq_{\\mathcal{I}}\\cap (\\mathcal{D}_{Fin}\\times \\mathcal{D}_{Fin}))$ describing the smallest sizes of unbounded from below with respect to the order $\\leq_{\\mathcal{I}}$ sets in $\\mathcal{D}_{Fin}$ and $\\mathcal{D}_{\\mathcal{I}}$, respectively. For a maximal ideal $\\mathcal{I}$, these cardinals were investigated by M. Canjar in connection with coinitial and cofinal subsets of the ultrapowers. We show that $\\mathfrak{b}(\\geq_{\\mathcal{I}}\\cap (\\mathcal{D}_{Fin} \\times \\mathcal{D}_{Fin})) =\\mathfrak{b}$ for all ideals $\\mathcal{I}$ with the Baire property and that $\\aleph_1 \\leq \\mathfrak{b}(\\geq_{\\mathcal{I}}\\cap (\\mathcal{D}_{\\mathcal{I}} \\times \\mathcal{D}_{\\mathcal{I}})) \\leq\\mathfrak{b}$ for all coanalytic weak P-ideals (this class contains all $\\Pi^0_4$ ideals). What is more, we give examples of Borel (even $\\Sigma^0_2$) ideals $\\mathcal{I}$ with $\\mathfrak{b}(\\geq_{\\mathcal{I}}\\cap (\\mathcal{D}_{\\mathcal{I}} \\times \\mathcal{D}_{\\mathcal{I}}))=\\mathfrak{b}$ as well as with $\\mathfrak{b}(\\geq_{\\mathcal{I}}\\cap (\\mathcal{D}_{\\mathcal{I}} \\times \\mathcal{D}_{\\mathcal{I}})) =\\aleph_1$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-07-29T16:09:30.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "03E05",
      "03E15",
      "03E17",
      "03E35"
    ],
    "keywords": [
      "ideal version",
      "bounding number",
      "coanalytic weak p-ideals",
      "finite-to-one functions",
      "cardinal numbers"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}