{
  "annotation_id": "90bf0a48-79fb-46f7-bb0c-080668438e7c",
  "date_created": "2026-02-17T05:53:26.417000Z",
  "date_modified": "2026-02-17T05:53:26.417000Z",
  "file_hash": "12a956831ec015351d7e93fe0f461e9a28d58d1531533cb8cc7e0f923ed7306e",
  "private": false,
  "record": {
    "abstract": "The manuscript is concerned with the Rudin-Keisler order of ultrafilters on measurable cardinals. The main theorem proved read as follows: Given regular cardinals $\\lambda\\leq \\kappa$, the following theories are equiconsistent modulo ZFC: (1) $\\kappa$ is a measurable cardinal with $o(\\kappa)=\\lambda^+$ (resp. $o(\\kappa)=\\kappa$). (2) The Rudin-Keisler order restricted to the set of $\\kappa$-complete (non-principal) ultrafilters on $\\kappa$ is $\\lambda^+$-directed (resp. $\\kappa^+$-directed). The theorem reported here is proved after bridging the directedness of the RK-order with the $\\lambda$-Gluing Property introduced by the authors in \\cite{HP}. Our result provides what seems to be the first example of a compactness-type property at the level of measurable cardinals whose consistency strength is much lower than the existence of a strong cardinal. As part of our analysis we also answer a question of Gitik by showing that the $\\aleph_0$-Gluing Property fails in his classical model from \u0027\u0027Changing cofinalities and the nonstationary ideal\". As a consequence of this, in Gitik\u0027s model the Rudin-Keisler order fails to be $\\aleph_1$-directed.",
    "arxiv_id": "2601.10614",
    "authors": [
      "Yair Hayut",
      "Alejandro Poveda"
    ],
    "categories": [
      "math.LO"
    ],
    "license": "http://creativecommons.org/licenses/by/4.0/",
    "title": "The directedness of the Rudin-Keisler order at measurable cardinals",
    "url": "https://arxiv.org/abs/2601.10614",
    "version": "v1"
  },
  "schema_id": "dorsal/arxiv",
  "source": {
    "execution_id": "d386619a-4735-4222-8696-456443cbf5d8",
    "id": "arXiv Dataset",
    "type": "Model",
    "variant": "snapshot-2026-01-17",
    "version": "0.1.0"
  },
  "user_id": 1000002
}