{
  "id": "1904.10420",
  "version": "v1",
  "published": "2019-04-23T16:55:59.000Z",
  "updated": "2019-04-23T16:55:59.000Z",
  "title": "Projection bands and atoms in pervasive pre-Riesz spaces",
  "authors": [
    "Anke Kalauch",
    "Helena Malinowski"
  ],
  "categories": [
    "math.FA"
  ],
  "abstract": "In vector lattices, the concept of a projection band is a basic tool. We deal with projection bands in the more general setting of an Archimedean pre-Riesz space $X$. We relate them to projection bands in a vector lattice cover $Y$ of $X$. If $X$ is pervasive, then a projection band in $X$ extends to a projection band in $Y$, whereas the restriction of a projection band $B$ in $Y$ is not a projection band in $X$, in general. We give conditions under which the restriction of $B$ is a projection band in $X$. We introduce atoms and discrete elements in $X$ and show that every atom is discrete. The converse implication is true, provided $X$ is pervasive. In this setting, we link atoms in $X$ to atoms in $Y$. If $X$ contains an atom $a>0$, we show that the principal band generated by $a$ is a projection band. Using atoms in a finite dimensional Archimedean pre-Riesz space $X$, we establish that $X$ is pervasive if and only if it is a vector lattice.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-04-23T16:55:59.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "46A40",
      "06F20",
      "47B65"
    ],
    "keywords": [
      "projection band",
      "pervasive pre-riesz spaces",
      "finite dimensional archimedean pre-riesz space",
      "vector lattice cover",
      "converse implication"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}