{
  "id": "1701.03915",
  "version": "v1",
  "published": "2017-01-14T12:57:23.000Z",
  "updated": "2017-01-14T12:57:23.000Z",
  "title": "Structures of lattices which can be represented as the collection of all up-sets",
  "authors": [
    "Peng He",
    "Xue-ping Wang"
  ],
  "comment": "15",
  "categories": [
    "math.RT"
  ],
  "abstract": "This paper first gives a necessary and sufficient condition that a lattice $L$ can be represented as the collection of all up-sets of a poset. Applying the condition, it obtains a necessary and sufficient condition that a lattice can be embedded into the lattice $L$ such that all infima, suprema, the top and bottom elements are preserved under the embedding by defining a monotonic operator on a poset. This paper finally shows that the quotient of the set of the monotonic operators under an equivalence relation can be naturally ordered and it is a lattice if $L$ is a finite distributive lattice.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-01-14T12:57:23.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "collection",
      "structures",
      "monotonic operator",
      "sufficient condition",
      "paper first"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}