Boolean Algebras and Stone Duality
The Stone Representation Theorem
Every Boolean algebra is isomorphic to a field of sets. The theorem, its proof from the prime ideal theorem, and what it does and does not deliver.
Learning objectives
- State the representation theorem in both forms
- Follow the proof via ultrafilters
- Distinguish the representation theorem from the full duality
Two forms of the statement
Every Boolean algebra is isomorphic to a field of sets — a subalgebra of some power set Su(X) under union, intersection and complement.
Every Boolean algebra B is isomorphic to the algebra of clopen subsets of its Stone space B*.
The second form is stronger: it names a canonical X and identifies the image exactly, rather than merely embedding into some power set.
The proof
- Take X to be the set of ultrafilters of B.
- Define σ(b) = {U : b ∈ U}.
- σ is a homomorphism. Because ultrafilters are closed under meet and decide complements, σ converts ∧ to intersection, ∨ to union, and ′ to complement.
- σ is injective. If a ≠ b then, without loss of generality, a ∧ b′ is non-zero. The filter it generates is proper and extends by BPI to an ultrafilter containing a but not b.
Steps 1–3 are routine. Step 4 requires that there be enough ultrafilters to separate points, which is precisely the Boolean prime ideal theorem. In ZF without choice the theorem can fail: there are models with Boolean algebras admitting no non-principal ultrafilters at all.
Relation to the subdirect representation
Stone's theorem is the concrete form of the fact that 2 is the only subdirectly irreducible Boolean algebra.
Given the subdirect irreducible analysis, the representation theorem is a corollary of Birkhoff's general theorem. What Stone's construction adds is canonicity of the index set and, subsequently, the topology.
What the theorem does not give
- The image is not all of Su(X). Only complete atomic algebras are full power sets. Generally σ(B) is a proper subalgebra.
- No control over X without topology. The bare representation theorem allows any sufficiently large index set. Stone's space is canonical only once the topology is imposed.
- Homomorphisms are not yet handled. The theorem represents objects but says nothing about maps. Turning it into a functorial correspondence is the content of the duality.
The gap between representation and duality is exactly the topology. Adding it makes the correspondence functorial and reversible, which is the subject of the next page.
Frequently asked questions
Is the representation theorem constructive?
No. It depends on BPI, which is not provable in ZF. For finite Boolean algebras the representation is constructive, since ultrafilters correspond to atoms.
Does an analogous theorem hold for distributive lattices?
Yes — Birkhoff's representation theorem for finite distributive lattices, and Priestley duality in the general case. Priestley duality adds an order to the topological space, since distributive lattices lack complementation.
Source. S. Burris and H. P. Sankappanavar, A Course in Universal Algebra, The Millennium Edition — a corrected re-typesetting of Springer-Verlag Graduate Texts in Mathematics 78 (1981). Section IV.4, book pages 152-156.
This page is an original exposition prepared for the KEVOS® knowledge library. It restates and reorganises mathematical results; it is not a reproduction of the source text.