{
  "id": "math/0604191",
  "version": "v1",
  "published": "2006-04-08T21:25:24.000Z",
  "updated": "2006-04-08T21:25:24.000Z",
  "title": "A connection between decomposability of ultrafilters and possible cofinalities",
  "authors": [
    "Paolo Lipparini"
  ],
  "comment": "8 pages",
  "categories": [
    "math.LO"
  ],
  "abstract": "We introduce the decomposability spectrum $K_D=\\{\\lambda \\geq \\omega| D \\text{is} \\lambda\\text{-decomposable}\\}$ of an ultrafilter $D$, and show that Shelah's $\\pcf$ theory influences the possible values $K_D$ can take. For example, we show that if $\\aaa$ is a set of regular cardinals, $\\mu \\in \\pcfa$, the ultrafilter $D$ is $|\\aaa |^+$-complete and $K_D \\subseteq \\aaa$, then $\\mu \\in K_D$. As a consequence, we show that if $ \\lambda $ is singular and for some $ \\lambda' < \\lambda $ $K_D$ contains all regular cardinals in $ [\\lambda', \\lambda)$ then: (a) if $\\cf \\lambda = \\omega $ then either $ \\lambda \\in K_D$, or $ \\lambda ^+ \\in K_D$; and (b) if $D$ is $(\\cf \\lambda)^+$-complete then $ \\lambda ^+ \\in K_D$, and $\\pp (\\lambda)= \\lambda ^+$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2006-04-08T21:25:24.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "03E04"
    ],
    "keywords": [
      "ultrafilter",
      "connection",
      "cofinalities",
      "regular cardinals"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 8,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2006math......4191L"
    }
  }
}