{
  "id": "2103.01581",
  "version": "v1",
  "published": "2021-03-02T09:15:31.000Z",
  "updated": "2021-03-02T09:15:31.000Z",
  "title": "Resolutions of Convex Geometries",
  "authors": [
    "Domenico Cantone",
    "Jean-Paul Doignon",
    "Alfio Giarlotta",
    "Stephen Watson"
  ],
  "comment": "Submitted",
  "categories": [
    "math.CO"
  ],
  "abstract": "Convex geometries (Edelman and Jamison, 1985) are finite combinatorial structures dual to union-closed antimatroids or learning spaces. We define an operation of resolution for convex geometries, which replaces each element of a base convex geometry by a fiber convex geometry. Contrary to what happens for similar constructions -- compounds of hypergraphs, as in Chein, Habib and Maurer (1981), and compositions of set systems, as in Mohring and Radermacher (1984) -- , resolutions of convex geometries always yield a convex geometry. We investigate resolutions of special convex geometries: ordinal and affine. A resolution of ordinal convex geometries is again ordinal, but a resolution of affine convex geometries may fail to be affine. A notion of primitivity, which generalize the corresponding notion for posets, arises from resolutions: a convex geometry is primitive if it is not a resolution of smaller ones. We obtain a characterization of affine convex geometries that are primitive, and compute the number of primitive convex geometries on at most four elements. Several open problems are listed.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-03-02T09:15:31.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05B35",
      "52B40"
    ],
    "keywords": [
      "resolution",
      "affine convex geometries",
      "finite combinatorial structures dual",
      "ordinal convex geometries",
      "base convex geometry"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}