{
  "id": "math/9204210",
  "version": "v1",
  "published": "1992-04-23T00:00:00.000Z",
  "updated": "1992-04-23T00:00:00.000Z",
  "title": "Reaping Numbers of Boolean Algebras",
  "authors": [
    "A. Dow",
    "J Steprāns",
    "W. S. Watson"
  ],
  "categories": [
    "math.LO"
  ],
  "abstract": "A subset $A$ of a Boolean algebra $B$ is said to be $(n,m)$-reaped if there is a partition of unity $P \\subset B$ of size $n$ such that the cardinality of $\\{b \\in P: b \\wedge a \\neq \\emptyset\\}$ is greater than or equal to $m$ for all $a\\in A$. The reaping number $r_{n,m}(B)$ of a Boolean algebra $B$ is the minimum cardinality of a set $A \\subset B\\setminus \\{0\\}$ such which cannot be $(n,m)$-reaped. It is shown that, for each $n \\in \\omega$, there is a Boolean algebra $B$ such that $r_{n+1,2}(B) \\neq r_{n,2}(B)$. Also, $\\{r_{n,m}(B) : \\{n,m\\}\\subseteq\\omega\\}$ consists of at most two consecutive integers. The existence of a Boolean algebra $B$ such that $r_{n,m}(B) \\neq r_{n',m'}(B)$ is equivalent to a statement in finite combinatorics which is also discussed.",
  "revisions": [
    {
      "version": "v1",
      "updated": "1992-04-23T00:00:00.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "boolean algebra",
      "reaping number",
      "minimum cardinality",
      "finite combinatorics",
      "consecutive integers"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "1992math......4210D"
    }
  }
}