{
  "id": "1908.06897",
  "version": "v1",
  "published": "2019-08-19T15:56:19.000Z",
  "updated": "2019-08-19T15:56:19.000Z",
  "title": "Strong G-schemes and strict homomorphisms",
  "authors": [
    "Frank a Campo"
  ],
  "comment": "23 pages, 9 figures",
  "categories": [
    "math.CO"
  ],
  "abstract": "Let $\\mathfrak{P}_r$ be a representation system of the non-isomorphic finite posets, and let ${\\cal H}(P,Q)$ be the set of order homomorphisms from $P$ to $Q$. For finite posets $R$ and $S$, we write $R \\sqsubseteq_G S$ iff, for every $P \\in \\mathfrak{P}_r$, a one-to-one mapping $\\rho_P : {\\cal H}(P,R) \\rightarrow {\\cal H}(P,S)$ exists which fulfills a certain regularity condition. It is shown that $R \\sqsubseteq_G S$ is equivalent to $\\# {\\cal S}(P,R) \\leq \\# {\\cal S}(P,S)$ for every finite posets $P$, where ${\\cal S}(P,Q)$ is the set of strict order homomorphisms from $P$ to $Q$. In consequence, $\\# {\\cal S}(P,R) = \\# {\\cal S}(P,S)$ holds for every finite posets $P$ iff $R$ and $S$ are isomorphic. A sufficient condition is derived for $R \\sqsubseteq_G S$ which needs the inspection of a finite number of posets only. Additionally, a method is developed which facilitates for posets $P + Q$ (direct sum) the construction of posets $T$ with $P + Q \\sqsubseteq_G A + T$, where $A$ is a convex subposet of $P$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-08-19T15:56:19.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "06A07",
      "06A06"
    ],
    "keywords": [
      "strong g-schemes",
      "strict homomorphisms",
      "non-isomorphic finite posets",
      "strict order homomorphisms",
      "convex subposet"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 23,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}