representing a Boolean algebra by field of sets

In this entry, we show that every Boolean algebraMathworldPlanetmath is isomorphic to a field of sets, originally noted by Stone in 1936. The bulk of the proof has actually been carried out in this entry (, which we briefly state:

if L is a distributive latticeMathworldPlanetmath, and X the set of all prime idealsMathworldPlanetmathPlanetmathPlanetmath of L, then the map F:LP(X) defined by F(a)={PaP} is an embeddingMathworldPlanetmathPlanetmath.

Now, if L is a Boolean lattice, then every element aL has a complementPlanetmathPlanetmath aL. a is in fact uniquely determined by a.

Proposition 1.

The embedding F above preserves in the following sense:


PF(a) iff aP iff aP iff PF(a) iff PX-F(a). ∎

Theorem 1.

Every Boolean algebra is isomorphic to a field of sets.


From what has been discussed so far, F is a Boolean algebra isomorphismMathworldPlanetmathPlanetmathPlanetmath between L and F(L), which is a ring of sets first of all, and a field of sets, because X-F(a)=F(a). ∎

Remark. There are at least two other ways to characterize a Boolean algebra as a field of sets: let L be a Boolean algebra:

  • Every prime ideal is the kernel of a homomorphismMathworldPlanetmath into 𝟐:={0,1}, and vice versa. So for an element a to be not in a prime ideal P is the same as saying that ϕ(a)=1 for some homomorphism ϕ:L𝟐. If we take Y to be the set of all homomorphisms from L to 𝟐, and define G:LP(Y) by G(a)={ϕϕ(a)=1}, then it is easy to see that G is an embedding of L into P(Y).

  • Every prime ideal is a maximal idealMathworldPlanetmath, and vice versa. Furthermore, P is maximal iff P is an ultrafilterMathworldPlanetmathPlanetmath. So if we define Z to be the set of all ultrafilters of L, and set H:LP(Z) by H(a)={UaU}, then it is easy to see that H is an embedding of L into P(Z).

If we appropriately topologize the sets X,Y, or Z, then we have the content of the Stone representation theorem.

Title representing a Boolean algebra by field of sets
Canonical name RepresentingABooleanAlgebraByFieldOfSets
Date of creation 2013-03-22 19:08:27
Last modified on 2013-03-22 19:08:27
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 13
Author CWoo (3771)
Entry type Theorem
Classification msc 06E20
Classification msc 06E05
Classification msc 03G05
Classification msc 06B20
Classification msc 03G10
Related topic FieldOfSets
Related topic RepresentingADistributiveLatticeByRingOfSets
Related topic LatticeHomomorphism
Related topic RepresentingACompleteAtomicBooleanAlgebraByPowerSet
Related topic StoneRepresentationTheorem
Related topic MHStonesRepresentationTheorem