Boolean Algebras and Stone Duality
Stone Duality and Boolean Spaces
Boolean algebras and Boolean spaces are the same theory read forwards and backwards. Stone duality is the template every later representation theorem in the subject imitates.
- Construct the Stone space of a Boolean algebra.
- Define a Boolean space and verify the Stone space is one.
- State the duality in both directions and as a categorical equivalence.
- Trace how algebraic notions correspond to topological ones.
- Recognise Cantor space as the dual of the countable atomless algebra.
- Explain why the duality is the template for Boolean products.
01The Stone space
Given a Boolean algebra B, take as points the ultrafilters on B, and topologise by declaring the sets of ultrafilters containing a given element to be a basis.
- The basis behaves algebraically[a ∧ b] = [a] ∩ [b], [a ∨ b] = [a] ∪ [b], [a′] = S(B) \ [a], [0] = ∅, [1] = S(B). The map a ↦ [a] is a Boolean homomorphism.
- It is injectiveIf a ≠ b then a △ b ≠ 0, and by BPI some ultrafilter contains a △ b, separating [a] from [b]. This step is exactly where BPI enters.
- Each [a] is clopenOpen by definition; closed because its complement is [a′], also basic open.
- The image is all clopen setsCompactness forces any clopen set to be a finite union of basic sets, hence of the form [a].
02Boolean spaces
A Boolean space — also called a Stone space or a profinite space — is a compact Hausdorff totally disconnected topological space.
| Condition | Algebraic counterpart |
|---|---|
| Compact | every element of B is a finite join; ultrafilters decide consistently |
| Hausdorff | distinct ultrafilters are separated by some element |
| Totally disconnected | the clopen sets form a basis, so B is recovered from them |
For every Boolean algebra B, the Stone space S(B) is a Boolean space, and B is isomorphic to the algebra of clopen subsets of S(B). Conversely, for every Boolean space X, the clopen sets form a Boolean algebra whose Stone space is homeomorphic to X.
Compactness of S(B) is exactly the finite intersection property argument: a family of basic closed sets with the FIP generates a filter, which extends to an ultrafilter lying in all of them. So compactness of the space and the ultrafilter extension principle are the same fact wearing different clothes.
03The duality as a correspondence
The two constructions are mutually inverse and extend to maps, giving a dual equivalence between the category of Boolean algebras with homomorphisms and the category of Boolean spaces with continuous maps.
| Boolean algebra | Boolean space |
|---|---|
| Algebra B | Stone space S(B) of ultrafilters |
| Element a ∈ B | clopen set [a] |
| Homomorphism B → C | continuous map S(C) → S(B) (direction reverses) |
| Injective homomorphism | surjective continuous map |
| Surjective homomorphism | embedding of a closed subspace |
| Ideal of B | open subset of S(B) |
| Filter of B | closed subset of S(B) |
| Ultrafilter | point |
| Quotient B/I | closed subspace |
| Finite algebra with n atoms | discrete space with n points |
| Atom of B | isolated point of S(B) |
| Atomless algebra | space with no isolated points |
| Direct product of algebras | disjoint union of spaces |
Reversal of direction is the hallmark of a duality rather than an equivalence, and it has real consequences: subobjects on one side correspond to quotients on the other, so the subalgebra lattice of B is the lattice of closed quotient spaces of S(B).
04Cantor space and the atomless case
The countable atomless Boolean algebra is unique up to isomorphism, and its Stone space is the Cantor set.
The absence of atoms corresponds exactly to the absence of isolated points. This is a good test of the dictionary: an algebraic condition stated in terms of the order translates into a purely topological one with no arithmetic in sight.
Questions that are awkward algebraically often become routine topologically and conversely. Deciding whether two Boolean algebras are isomorphic becomes deciding whether two spaces are homeomorphic; the classification of countable Boolean algebras is a topological classification of countable Boolean spaces.
05The template for later representations
Stone duality is the model that the rest of Chapter IV generalises. Each successive construction transports Boolean structure into a different variety.
- Stone, 1936Boolean algebras as clopen algebrasThe original duality. Every Boolean algebra is a field of sets, canonically.
- Arens and Kaplansky, 1948Filtered Boolean powersApplied to finite fields, transporting Boolean structure into ring theory.
- Boolean powersA[B] constructionsTake an algebra A and a Boolean algebra B and build a new algebra whose behaviour is A-like locally and B-like globally.
- Comer, 1970sSheaves over Boolean spacesRecognising the general pattern as a sheaf construction over the Stone space.
- Burris and Werner, 1979Boolean productsA simplification of Boolean sheaves that avoids the machinery while keeping the content. The formulation adopted by the source.
Every one of these is the same idea: index a family of algebras by the points of a Boolean space and require the assignment to vary continuously. The Boolean Constructions stream develops the whole sequence.
06What the duality does not do
Stone duality is exact for Boolean algebras and does not extend unchanged to nearby varieties.
The lesson is that the duality is a consequence of the very strong structure of Boolean algebras — one subdirectly irreducible, arithmetical, complemented — rather than a general phenomenon. Where those hypotheses weaken, so does the correspondence.
Frequently asked
Does Stone duality need the axiom of choice?
It needs BPI, which follows from AC and is strictly weaker. Without BPI the Stone space of an atomless Boolean algebra could be empty and the representation would fail. For finite Boolean algebras no choice is required at all — the duality reduces to the correspondence between a finite algebra and its atom set.
What is the Stone space of a power set algebra?
The Stone space of the power set of I is the Stone–Čech compactification of I with the discrete topology. Its points are the ultrafilters on I: the principal ones form a dense copy of I, and the free ones make up the remainder. It is compact but very far from metrisable when I is infinite.
Why does the arrow direction reverse?
Because the correspondence is contravariant: a homomorphism B → C pulls ultrafilters on C back to ultrafilters on B, giving a continuous map S(C) → S(B). This is the same reversal seen when passing from a ring to its spectrum, and it is what makes the relationship a duality rather than an equivalence.
- S. Burris and H. P. Sankappanavar, A Course in Universal Algebra, Millennium Edition (a corrected re-typesetting of Springer GTM 78, 1981).
- G. Grätzer, Universal Algebra, 2nd edition, Springer.
- R. McKenzie, G. McNulty and W. Taylor, Algebras, Lattices, Varieties, Volume I.
Original KEVOS® explanatory article. Written from the topic map of the cited works; no text is reproduced from them.