Boolean Algebras and Stone Duality
Maximal Filters and Boolean Congruences
The lattice isomorphism between filters and congruences, what maximal filters say about simple quotients, and the resulting proof that 2 is the only subdirectly irreducible Boolean algebra.
Learning objectives
- State the filter–congruence isomorphism precisely
- Show that maximal filters give simple quotients
- Derive that 2 is the unique subdirectly irreducible Boolean algebra
The isomorphism
For a Boolean algebra B, the map F ↦ θF is a lattice isomorphism from the lattice of filters of B onto Con B.
| Filter | Congruence | Quotient |
|---|---|---|
| {1} | Δ | B |
| B (improper) | ∇ | Trivial |
| An ultrafilter | A maximal congruence | 2 |
| A principal filter above a | Θ determined by a | The relative algebra below a′ |
Because the correspondence is a lattice isomorphism, every question about Boolean congruences is a question about filters. Filters are far more concrete, which is why the Boolean theory is so much more tractable than general lattice theory.
Maximal filters and simple quotients
By the correspondence theorem, congruences of B/F correspond to filters above F. Maximality of F means there are only two such filters, so the quotient has only two congruences and is simple. The two-element characterisation of ultrafilters identifies the quotient.
2 is the only subdirectly irreducible
- Suppose B has more than two elements, so there is a with 0 < a < 1.
- The principal filters generated by a and by a′ are both proper and distinct from {1}.
- Their corresponding congruences are both non-trivial, and their meet is Δ because the filters intersect in {1}.
- So Con B has two distinct atoms below which nothing lies in common — there is no monolith, and B is not subdirectly irreducible.
- Every Boolean algebra is a subdirect power of 2 — by Birkhoff's subdirect representation theorem.
- Every Boolean algebra embeds in a power set — the Stone representation theorem, since a subdirect power of 2 is a subalgebra of 2I ≅ Su(I).
- An identity holds in all Boolean algebras exactly when it holds in 2 — truth tables suffice.
- The variety is semisimple: every subdirectly irreducible member is simple.
Where the index set comes from
Birkhoff's theorem gives a subdirect representation but does not say what the index set is. For Boolean algebras it can be identified: the index set is the set of ultrafilters, and the embedding sends b to the set of ultrafilters containing it.
Making the index set canonical, and then topologising it, is what turns the representation theorem into the duality of the next pages.
Frequently asked questions
Does the filter–congruence isomorphism hold for distributive lattices?
Not in general. It relies on complementation and on congruence permutability. Distributive lattices are congruence-distributive but not permutable, so their congruences are not determined by a single filter.
Why is the subdirect representation not usually a direct product?
Because the embedding into the power of 2 is rarely surjective. Only complete atomic Boolean algebras are full power sets; the rest are proper subalgebras.
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 149-152.
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.