{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "B9T33D5MJK"
    }
  ],
  "body": "\n## Statement\n\nLet $B$ be a [[Boolean algebra|0XADJKM699]], and let $S(B)$ be the set of [[ultrafilters|CQM84EW10G]] of $B$.\nGive $S(B)$ the topology with basis the sets $\\hat a = \\set{U \\in S(B) : a \\in U}$ for $a \\in B$.\nThen $S(B)$ is a [[Stone space|ZV38HA5PAQ]].\nThe map $a \\mapsto \\hat a$ is an isomorphism from $B$ onto the Boolean algebra of [[clopen|DQ4E55T3BW]] subsets of $S(B)$.\nIn particular, every Boolean algebra is isomorphic to a [[field of sets|PFMGZ7988V]].\n\n## Remarks\n\nThe map $a \\mapsto \\hat a$ preserves the operations because an ultrafilter contains $a \\wedge b$ exactly when it contains $a$ and $b$, and contains $\\neg a$ exactly when it does not contain $a$.\nIt is injective by the [[ultrafilter lemma|3MKZST79MK]]: if $a \\ne b$, then $a \\wedge \\neg b \\ne 0$ or $\\neg a \\wedge b \\ne 0$, and an ultrafilter containing that element separates $a$ and $b$.\nCompactness of $S(B)$ also uses the ultrafilter lemma, and surjectivity onto the clopen sets follows from compactness, since a clopen set is a finite union of basic sets $\\hat a$.\nThe ultrafilter lemma cannot be dropped: over ZF, the theorem implies it.\n\nExample: let $B$ be the algebra of finite and cofinite subsets of $\\N$.\nIts ultrafilters are the principal ones and one more, made of the cofinite sets, and $S(B)$ is the one-point compactification of $\\N$.\nFor a finite Boolean algebra the space is finite and discrete, which gives [[Stone's representation theorem for finite Boolean algebras|0DYS9C3HH5]].\n\n## Explanations\n\n### L1 human\n\nEvery Boolean algebra can be pictured as a family of subsets of some set, with union, intersection and complement as its operations.\nThe set to use is the set of ultrafilters, the consistent ways to declare each element true or false.\nSo the axioms of a Boolean algebra say nothing beyond what holds for sets.\n\n### L2 human\n\nThe theorem is the object part of [[Stone duality|B9T33D5MJK]], which also matches homomorphisms with continuous maps.\nIt implies that an equation holds in all Boolean algebras as soon as it holds in the two-element algebra.\nThis is the completeness theorem for classical propositional logic in algebraic form.\nThe same idea, with prime filters instead of ultrafilters, gives [[Priestley duality|SA22SEC75K]] for distributive lattices.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "about": [
      "0XADJKM699",
      "ZV38HA5PAQ"
    ],
    "kind": "statement",
    "msc": [
      "06E15",
      "06E05"
    ],
    "name": "Stone's representation theorem for Boolean algebras",
    "role": "theorem",
    "same_as": {
      "wikidata": "Q1166774",
      "wikipedia": "Stone's representation theorem for Boolean algebras"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Stone%27s_representation_theorem_for_Boolean_algebras"
      }
    ],
    "tag": "S4QJ395PA5",
    "uses": [
      "0XADJKM699",
      "CQM84EW10G",
      "ZV38HA5PAQ",
      "DQ4E55T3BW",
      "PFMGZ7988V"
    ]
  },
  "kind": "statement",
  "link": "https://tamnd.github.io/math2/node/S4QJ395PA5/",
  "links": [
    {
      "field": "about",
      "target": "0XADJKM699"
    },
    {
      "field": "about",
      "target": "ZV38HA5PAQ"
    },
    {
      "field": "uses",
      "target": "0XADJKM699"
    },
    {
      "field": "uses",
      "target": "CQM84EW10G"
    },
    {
      "field": "uses",
      "target": "ZV38HA5PAQ"
    },
    {
      "field": "uses",
      "target": "DQ4E55T3BW"
    },
    {
      "field": "uses",
      "target": "PFMGZ7988V"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/S4QJ395PA5.md",
  "msc": [
    "06E15",
    "06E05"
  ],
  "name": "Stone's representation theorem for Boolean algebras",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/S4Q/S4QJ395PA5.md",
  "status": null,
  "tag": "S4QJ395PA5",
  "uri": "m2://S4QJ395PA5",
  "url": "https://tamnd.github.io/math2/wiki/stones-representation-theorem-for-boolean-algebras/",
  "writing": {
    "tag": "S4QJ395PA5",
    "kind": "statement",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      },
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}