{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $B$ be a [[Boolean algebra|0XADJKM699]] with order $\\le$.\nThe algebra $B$ is *complete* if every subset $S \\subseteq B$ has a supremum $\\bigvee S$ in $B$, that is, a least upper bound for $\\le$.\n\n## Remarks\n\nThen every subset also has an infimum, $\\bigwedge S = \\neg \\bigvee \\set{\\neg s : s \\in S}$, so $B$ is a [[complete lattice|D7A0ZGZT5S]].\nThe infinite distributive laws $a \\wedge \\bigvee S = \\bigvee \\set{a \\wedge s : s \\in S}$ hold in every complete Boolean algebra.\n\nExample: the power set of a set $X$, ordered by inclusion.\nThe supremum of a family of subsets is its union, which is again a subset of $X$.\n\nNon-example: the algebra $B$ of finite and cofinite subsets of $\\N$ is not complete.\nLet $S = \\set{\\set{2n} : n \\in \\N}$.\nAn upper bound of $S$ in $B$ contains all even numbers, so it is infinite and hence cofinite, and it contains infinitely many odd numbers.\nRemoving one odd number gives a smaller upper bound in $B$, so there is no least one.\n\nIn Mathlib the notion is `CompleteBooleanAlgebra`.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "75BPV95AAS",
    "kind": "definition",
    "msc": [
      "06E10"
    ],
    "name": "Complete Boolean algebra",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Complete_Boolean_algebra"
      }
    ],
    "tag": "VFKNNPVMTX",
    "uses": [
      "0XADJKM699",
      "D7A0ZGZT5S"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/VFKNNPVMTX/",
  "links": [
    {
      "field": "defines",
      "target": "75BPV95AAS"
    },
    {
      "field": "uses",
      "target": "0XADJKM699"
    },
    {
      "field": "uses",
      "target": "D7A0ZGZT5S"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/VFKNNPVMTX.md",
  "msc": [
    "06E10"
  ],
  "name": "Complete Boolean algebra",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/VFK/VFKNNPVMTX.md",
  "status": null,
  "tag": "VFKNNPVMTX",
  "uri": "m2://VFKNNPVMTX",
  "url": "https://tamnd.github.io/math2/wiki/complete-boolean-algebra-definition/",
  "writing": {
    "tag": "VFKNNPVMTX",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}