{
  "id": "1704.02494",
  "version": "v1",
  "published": "2017-04-08T13:46:09.000Z",
  "updated": "2017-04-08T13:46:09.000Z",
  "title": "Ideals in $\\mathcal{P} _G$ and $βG$",
  "authors": [
    "Igor Protasov",
    "Ksenia Protasova"
  ],
  "comment": "Keywords: Stone-$\\check{C}$ech compactification, Boolean algebra, ideal, filter, ultrafilter",
  "categories": [
    "math.GN"
  ],
  "abstract": "For a discrete group $G$, we use the natural correspondence between ideals in the Boolean algebra $ \\mathcal{P}_G$ of subsets of $G$ and closed subsets in the Stone-$\\check{C}$ech compactifi-cation $\\beta G$ as a right topological semigroup to introduce and characterize some new ideals in $\\beta G$. We show that if a group $G$ is either countable or Abelian then there are no closed ideals in $\\beta G$ maximal in $G^*$, $G^* = \\beta G \\setminus G$, but this statement does not hold for the group $S_\\kappa$ of all permutations of an infinite cardinal $\\kappa$. We characterize the minimal closed ideal in $\\beta G$ containing all idempotents of $G^*$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-04-08T13:46:09.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "boolean algebra",
      "right topological semigroup",
      "discrete group",
      "infinite cardinal",
      "minimal closed ideal"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}