{
  "id": "2211.11192",
  "version": "v1",
  "published": "2022-11-21T05:48:21.000Z",
  "updated": "2022-11-21T05:48:21.000Z",
  "title": "Characterizations of the projection bands and some order properties of the space of continuous functions",
  "authors": [
    "Eugene Bilokopytov"
  ],
  "comment": "19 pages, preliminary version",
  "categories": [
    "math.FA",
    "math.GN"
  ],
  "abstract": "We show that for an ideal $H$ in an Archimedean vector lattice $F$ the following conditions are equivalent: $\\bullet$ $H$ is a projection band; $\\bullet$ Any collection of mutually disjoint vectors in $H$, which is order bounded in $F$, is order bounded in $H$; $\\bullet$ $H$ is an infinite meet-distributive element of the lattice $\\mathcal{I}_{F}$ of all ideals in $F$ in the sense that $\\bigcap\\limits_{J\\in \\mathcal{J}}\\left(H+ J\\right)=H+ \\bigcap \\mathcal{J}$, for any $\\mathcal{J}\\subset \\mathcal{I}_{F}$. Additionally, we show that if $F$ is uniformly complete and $H$ is a uniformly closed principal ideal, then $H$ is a projection band. In the process we investigate some order properties of lattices of continuous functions on Tychonoff topological spaces.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-11-21T05:48:21.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "06E15",
      "46A40",
      "46E25"
    ],
    "keywords": [
      "projection band",
      "order properties",
      "continuous functions",
      "characterizations",
      "archimedean vector lattice"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}