{
  "id": "1209.6444",
  "version": "v3",
  "published": "2012-09-28T08:01:09.000Z",
  "updated": "2014-12-03T18:11:52.000Z",
  "title": "Large free sets in universal algebras",
  "authors": [
    "Taras Banakh",
    "Artur Bartoszewicz",
    "Szymon Głab"
  ],
  "comment": "4 pages",
  "journal": "Algebra Universalis, 71 (2014) 23-29",
  "doi": "10.1007/s00012-013-0261-0",
  "categories": [
    "math.LO",
    "math.RA"
  ],
  "abstract": "We prove that for each universal algebra $(A,\\mathcal A)$ of cardinality $|A|\\ge 2$ and an infinite set $X$ of cardinality $|X|\\ge|\\mathcal A|$, the $X$-th power $(A^X,\\mathcal A^X)$ of the algebra $(A,\\mathcal A)$ contains a free subset $\\mathcal F\\subset A^X$ of cardinality $|\\mathcal F|=2^{|X|}$. This generalizes the classical Fichtenholtz-Kantorovitch-Hausdorff result on the existence of an independent family $\\mathcal I\\subset\\mathcal P(X)$ of cardinality $|\\mathcal I|=|\\mathcal P(X)|$ in the Boolean algebra $\\mathcal P(X)$ of subsets of an infinite set $X$.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2013-01-09T13:24:34.000Z",
      "title": "Independent and free sets in universal algebras",
      "abstract": "We discuss notions of independent and free sets in universal algebras, and more generally, in spaces endowed with a hull operator. Our main result says that for each universal algebra $\\mathbb A=(A,\\mathcal A)$ of cardinality $|A|\\ge 2$ and an infinite set $X$ of cardinality $|X|\\ge|\\mathcal A|$, the $X$-power $\\mathbb A^X=(A^X,\\mathcal A^X)$ of the algebra $\\mathbb A$ contains an $\\mathcal A$-free subset $\\mathcal F\\subset A^X$ of cardinality $|\\mathcal F|=2^{|X|}$. This generalizes a classical Fichtenholtz-Katorovitch-Hausdorff result on the existence of large independent families of subsets of a given infinite set.",
      "comment": "10 pages",
      "journal": null,
      "doi": null
    },
    {
      "version": "v3",
      "updated": "2014-12-03T18:11:52.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05B35",
      "08A99",
      "17A50"
    ],
    "keywords": [
      "universal algebra",
      "free sets",
      "infinite set",
      "main result says",
      "large independent families"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 4,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1209.6444B"
    }
  }
}