---
tag: S4QJ395PA5
kind: statement
role: theorem
name: Stone's representation theorem for Boolean algebras
msc: ["06E15", "06E05"]
uses: [0XADJKM699, CQM84EW10G, ZV38HA5PAQ, DQ4E55T3BW, PFMGZ7988V]
about: [0XADJKM699, ZV38HA5PAQ]
sources:
  - url: https://en.wikipedia.org/wiki/Stone%27s_representation_theorem_for_Boolean_algebras
same_as:
  wikidata: Q1166774
  wikipedia: Stone's representation theorem for Boolean algebras
---

## Statement

Let $B$ be a [[Boolean algebra|0XADJKM699]], and let $S(B)$ be the set of [[ultrafilters|CQM84EW10G]] of $B$.
Give $S(B)$ the topology with basis the sets $\hat a = \set{U \in S(B) : a \in U}$ for $a \in B$.
Then $S(B)$ is a [[Stone space|ZV38HA5PAQ]].
The map $a \mapsto \hat a$ is an isomorphism from $B$ onto the Boolean algebra of [[clopen|DQ4E55T3BW]] subsets of $S(B)$.
In particular, every Boolean algebra is isomorphic to a [[field of sets|PFMGZ7988V]].

## Remarks

The 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 $a$.
It 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 $b$.
Compactness 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 $\hat a$.
The ultrafilter lemma cannot be dropped: over ZF, the theorem implies it.

Example: let $B$ be the algebra of finite and cofinite subsets of $\N$.
Its ultrafilters are the principal ones and one more, made of the cofinite sets, and $S(B)$ is the one-point compactification of $\N$.
For a finite Boolean algebra the space is finite and discrete, which gives [[Stone's representation theorem for finite Boolean algebras|0DYS9C3HH5]].

## Explanations

### L1 human

Every Boolean algebra can be pictured as a family of subsets of some set, with union, intersection and complement as its operations.
The set to use is the set of ultrafilters, the consistent ways to declare each element true or false.
So the axioms of a Boolean algebra say nothing beyond what holds for sets.

### L2 human

The theorem is the object part of [[Stone duality|B9T33D5MJK]], which also matches homomorphisms with continuous maps.
It implies that an equation holds in all Boolean algebras as soon as it holds in the two-element algebra.
This is the completeness theorem for classical propositional logic in algebraic form.
The same idea, with prime filters instead of ultrafilters, gives [[Priestley duality|SA22SEC75K]] for distributive lattices.