Boolean Algebras and Stone Duality
Filters, Ideals and Ultrafilters
An ultrafilter is a consistent way of deciding every question at once. They are the points of the Stone space, the indices of ultraproducts, and the reason compactness works.
- Define filters and ideals and describe the duality between them.
- Distinguish principal from free filters.
- State the three equivalent characterisations of an ultrafilter.
- Construct the filter generated by a set with the finite intersection property.
- Explain why free ultrafilters require a choice principle.
- Preview the role of ultrafilters in ultraproducts and Stone duality.
01Filters and ideals
A filter on a Boolean algebra B is a non-empty subset F closed upward and closed under meet; it is proper when 0 ∉ F. An ideal is the order dual.
| Filter F | Ideal I | |
|---|---|---|
| Contains | 1 | 0 |
| Closed under | finite meets | finite joins |
| Closed | upward | downward |
| Proper when | 0 ∉ F | 1 ∉ I |
| Complement map | { x′ : x ∈ F } is an ideal | { x′ : x ∈ I } is a filter |
| Congruence class | the 1-class | the 0-class |
Complementation exchanges the two notions exactly, so any theorem about filters has a dual about ideals and neither needs separate proof. The lattice of filters is dually isomorphic to the lattice of ideals, and both are isomorphic to Con B.
02Principal and free filters
The principal filter generated by an element a is the set of elements above a. A filter is free when it is not principal.
The standard example of a free filter is the Fréchet filter on the power set of an infinite set: the cofinite subsets. It is a proper filter, its members have empty intersection, and it is contained in many free ultrafilters — though exhibiting even one of those needs Zorn's lemma.
03Ultrafilters
An ultrafilter is a maximal proper filter. Three characterisations coincide, and each is used in practice.
For a proper filter U on a Boolean algebra B, the following are equivalent: (i) U is maximal among proper filters; (ii) for every a ∈ B, exactly one of a and a′ lies in U; (iii) whenever a ∨ b ∈ U, either a ∈ U or b ∈ U.
- Maximality ⟹ decisivenessIf neither a nor a′ were in U, adjoining a would give a larger proper filter, contradicting maximality. Both cannot be in U since their meet is 0.
- Decisiveness ⟹ primenessIf a ∨ b ∈ U and a ∉ U then a′ ∈ U, so b ≥ (a ∨ b) ∧ a′ is in U by upward closure and meet closure.
- Primeness ⟹ maximalityA proper filter properly containing a prime filter would have to contain some a with a′ already inside, forcing 0 into the filter.
- The quotient readingU is an ultrafilter exactly when B/U ≅ 2 — the corresponding congruence is a coatom of Con B. This connects to maximal ideals in the ring picture.
Characterisation (ii) is the one to carry: an ultrafilter decides every question. For every element it commits to either that element or its complement, consistently. That is what makes ultrafilters usable as a device for taking limits.
04Existence and the finite intersection property
A family with the finite intersection property — every finite subfamily has non-zero meet — generates a proper filter, and Zorn's lemma extends it to an ultrafilter.
- input: family S ⊆ B with the finite intersection property
- F₀ := { x ∈ B : x ≥ s₁ ∧ ⋯ ∧ sₙ for some finite s₁,…,sₙ ∈ S }
- F₀ is a proper filter, since finite meets from S are non-zero
- consider the poset of proper filters containing F₀, ordered by inclusion
- every chain has an upper bound: the union, which is proper
- (0 lies in no member, so 0 lies in no union)
- by Zorn's lemma there is a maximal element U
- U is an ultrafilter containing S
The existence of a free ultrafilter on the natural numbers is not provable in ZF alone. It follows from the Boolean Prime Ideal Theorem, which is strictly weaker than the axiom of choice but not a theorem of ZF. Any argument claiming to construct one explicitly is mistaken, and any theorem relying on one carries that choice principle as a hypothesis.
05Ultrafilters on a set
The most-used case is B = the power set of a set I. Here an ultrafilter is a family of subsets of I deciding, for each subset, whether it is 'large'.
| Kind | Description | Exists? |
|---|---|---|
| Principal at i | all subsets containing the fixed point i | always, explicitly |
| Free | contains all cofinite sets, no finite set | only via BPI; never explicit |
| On a finite set | principal only | no free ultrafilters exist |
The distinction matters immediately for ultraproducts: an ultraproduct over a principal ultrafilter collapses to a single factor and gives nothing new, whereas an ultraproduct over a free ultrafilter is the construction that yields compactness, non-standard models and Jónsson's lemma. Every interesting ultraproduct uses a free ultrafilter.
06Where ultrafilters go next
Frequently asked
Is every filter contained in an ultrafilter?
Every proper filter is, by the Zorn's lemma argument. The improper filter — the whole algebra — is not, since ultrafilters are proper by definition. The extension result is exactly the Boolean Prime Ideal Theorem in filter form.
Why does an ultraproduct over a principal ultrafilter collapse?
Because the ultrafilter concentrates all its attention on a single index. Two elements of the product are identified when they agree on a member of the ultrafilter, and every member contains the distinguished point, so agreement at that one coordinate suffices. The ultraproduct is therefore isomorphic to that single factor.
Do ultrafilters exist on every Boolean algebra?
Proper ultrafilters exist on every non-trivial Boolean algebra, by extending the principal filter of any non-zero element. On the one-element algebra there are none, since every filter contains 0. The interesting question is whether free ultrafilters exist, which requires the algebra to be infinite and requires BPI.
- 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.