Boolean Algebras and Stone Duality
Ultrafilters and the Boolean Prime Ideal Theorem
Maximal proper filters, their characterisation by the decision property, and the existence theorem that underwrites Stone duality and the ultraproduct construction.
Learning objectives
- Define ultrafilter and give equivalent characterisations
- State the Boolean prime ideal theorem
- Distinguish principal from free ultrafilters
Ultrafilters
A proper filter U that is maximal among proper filters.
For a proper filter U of a Boolean algebra B, the following are equivalent:
- U is maximal among proper filters;
- for every a ∈ B, exactly one of a and a′ lies in U;
- a ∨ b ∈ U implies a ∈ U or b ∈ U;
- B/U is isomorphic to 2.
Condition (4) is the one that does the work: an ultrafilter is exactly the preimage of 1 under a homomorphism onto 2. So ultrafilters and homomorphisms to 2 are the same data, which is why they index the Stone space.
The existence theorem
Every proper filter of a Boolean algebra extends to an ultrafilter. Equivalently, every proper ideal extends to a maximal ideal.
The proof is Zorn's lemma applied to the proper filters containing the given one. The union of a chain of proper filters is a proper filter, since properness — omitting 0 — is preserved by unions of chains.
BPI is strictly weaker than the full axiom of choice but is not provable in ZF alone. It is equivalent to the compactness theorem for first-order logic, to Tychonoff's theorem for Hausdorff spaces, and to Stone's representation theorem. Several results in Chapters IV and V depend on it essentially.
Principal and free ultrafilters
One generated by a single atom: U = {b : b ≥ a} for an atom a.
One that is not principal. In a power set algebra, equivalently one containing the Fréchet filter of cofinite sets.
| Principal | Free | |
|---|---|---|
| Generated by | An atom | No single element |
| Existence | Constructive | Requires BPI |
| In a finite algebra | All ultrafilters are principal | None exist |
| Ultraproduct modulo it | Isomorphic to one factor | A genuinely new algebra |
| In Su(X) for infinite X | One per point of X | 22|X| of them |
An ultraproduct modulo a principal ultrafilter is just one of the factors, so the construction is trivial. All the content of the ultraproduct method — the compactness theorem, Jónsson's lemma, non-standard models — depends on free ultrafilters, and hence on BPI.
Ultrafilters in the finite case
In a finite Boolean algebra every ultrafilter is principal and corresponds to an atom, so ultrafilters are in bijection with atoms. Since finite Boolean algebras are power sets, this says ultrafilters on a finite set correspond to points.
The infinite case is where the theory becomes substantial: there are far more ultrafilters than points, and the Stone space of Su(N) — the &Cech–Stone compactification βN — has cardinality 22ℵ0.
Frequently asked questions
Can a free ultrafilter be described explicitly?
No. Their existence requires a choice principle, and it is consistent with ZF that none exists. No explicit definition is possible.
What does 'exactly one of a and a′' mean intuitively?
An ultrafilter decides every question. Thinking of the filter as a notion of 'large', an ultrafilter is a notion of largeness under which every set is either large or has large complement, with no middle case.
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.3, book pages 146-149.
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.