{
  "id": "1906.11758",
  "version": "v1",
  "published": "2019-06-27T16:00:47.000Z",
  "updated": "2019-06-27T16:00:47.000Z",
  "title": "About Generalized One-to-One Mappings between Sets of Order Homomorphisms",
  "authors": [
    "Frank a Campo"
  ],
  "comment": "35 pages, 9 figures",
  "categories": [
    "math.CO"
  ],
  "abstract": "Structural properties of finite posets $R$ and $S$ are studied which enforce $\\# H(P,R) \\leq \\# H(P,S)$ for every finite poset $P$, where $H(P,Q)$ is the set of order homomorphisms from $P$ to $Q$. The concept of the strong Hom-scheme is introduced. In the case of existence, a strong Hom-scheme from $R$ to $S$ defines a one-to-one mapping $\\rho_P : H(P,R) \\rightarrow H(P,S)$ for every poset $P \\in P_r$, where $P_r$ is a representation system of the non-isomorphic finite posets. By postulating regularity conditions for the way, how a strong Hom-scheme maps the elements of $H(P,R)$ to the elements of $H(P,S)$, the strong I-scheme from $R$ to $S$ is defined. The existence of a strong I-scheme from $R$ to $S$ turns out to be equivalent to the existence of a one-to-one homomorphism $\\epsilon$ between the so-called EV-systems of $R$ and $S$, where $\\epsilon$ has to fulfill an additional condition. Methods are developed which allow - in many cases - for given finite posets $R$ and $S$, the proof of the relation \"$\\# H(P,R) \\leq \\# H(P,S)$ for every finite poset $P$\" or the refutation of this relation.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-06-27T16:00:47.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "06A07",
      "06A06"
    ],
    "keywords": [
      "generalized one-to-one mappings",
      "order homomorphisms",
      "strong i-scheme",
      "non-isomorphic finite posets",
      "strong hom-scheme maps"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 35,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}