Boolean Algebras and Stone Duality

Boolean Algebras: Axioms and Structure

Q: Why is the two-element algebra so decisive?

Because it is the only subdirectly irreducible member, Birkhoff's subdirect representation theorem forces every Boolean algebra to be a subdirect power of it. No other information about the variety is needed — the single algebra determines everything, and questions about Boolean algebras reduce to questions about fields of sets.

Q: Are complemented distributive lattices the same as Boolean algebras?

The same objects, a different type. Complemented distributive lattices in type (2, 2) with existence axioms do not form a variety; promoting complementation to a unary operation gives the Boolean algebra type and does. Since complements are unique under distributivity, no object is gained or lost — only the subalgebra and homomorphism notions change.

Q: Is every infinite Boolean algebra a power set algebra?

No. Power set algebras are complete and atomic; the algebra of finite and cofinite subsets of an infinite set is neither. Every Boolean algebra embeds in a power set algebra, but the embedding is rarely onto, and characterising the image is exactly what Stone duality does.

Boolean algebras are the best-behaved non-trivial variety in the subject: arithmetical, locally finite, with exactly one subdirectly irreducible member. Almost everything else in Chapter IV is machinery for exporting that behaviour elsewhere.

Engineering · Mathematics6 min readKV-MATH-0232

Learning objectives

State the Boolean algebra axioms and identify the type.
Prove that the two-element algebra is the only subdirectly irreducible Boolean algebra.
Derive the subdirect representation of an arbitrary Boolean algebra.
Exhibit a Pixley term and conclude the variety is arithmetical.
Distinguish atomic from atomless Boolean algebras.
State the structure theorem for finite Boolean algebras.

01The variety

A Boolean algebra is an algebra of type (2, 2, 1, 0, 0) — join, meet, complement, zero and one — satisfying the distributive lattice axioms together with the complementation laws.

Axioms beyond the lattice identities
Group	Identities
Distributivity	x ∧ (y ∨ z) ≈ (x ∧ y) ∨ (x ∧ z) and its dual
Bounds	x ∨ 0 ≈ x, x ∧ 1 ≈ x
Complementation	x ∨ x′ ≈ 1, x ∧ x′ ≈ 0

Key resultComplement is an operation, not an existence claim

Placing complementation in the type is what makes the class a variety. If complements were merely asserted to exist, homomorphisms would not be obliged to preserve them and subalgebras would not be obliged to contain them. Because the distributive law forces complements to be unique when they exist, nothing is lost by promoting the existence claim to an operation.

Uniqueness of complements in a distributive lattice is a short argument: if b and c are both complements of a, then b = b ∧ 1 = b ∧ (a ∨ c) = (b ∧ a) ∨ (b ∧ c) = 0 ∨ (b ∧ c) = b ∧ c, so b ≤ c, and symmetry gives equality. In a non-distributive complemented lattice such as M5, complements genuinely fail to be unique.

02One subdirectly irreducible

The two-element Boolean algebra 2 is the only subdirectly irreducible member of the variety, and this single fact drives the rest of the theory.

ProcedureProving 2 is the only subdirectly irreducible Boolean algebra

in: Boolean algebra B, |B| > 2 → out: proof that B is subdirectly reducible

input: Boolean algebra B with more than two elements
choose a with 0 < a < 1
the pair {a, a′} of principal filters induces two congruences:
θ₁ := the congruence collapsing the interval [0, a]
θ₂ := the congruence collapsing the interval [0, a′]
θ₁ ∧ θ₂ = Δ (nothing is collapsed by both)
θ₁ ≠ Δ and θ₂ ≠ Δ (both collapse something, since 0 < a < 1)
so Δ is a meet of two non-trivial congruences
therefore B is not subdirectly irreducible
conclusion: only |B| = 2 survives

The two congruences correspond to the ideals generated by a and by a′, which are complementary. Correctness rests on complementation: in a general distributive lattice no such pair exists. Caveat: the one-element algebra is trivial and excluded by convention.

Combining with Birkhoff's subdirect representation theorem gives the classical result: every Boolean algebra is a subdirect power of 2, hence isomorphic to a field of sets. Stone duality, two pages on, sharpens this to a topological equivalence.

03The variety is arithmetical

Boolean algebras are congruence-permutable and congruence-distributive, hence arithmetical, and Pixley's theorem says a single ternary term witnesses this.

Pixley term: p(x, y, z) = (x ∧ z) ∨ (x ∧ y′) ∨ (y′ ∧ z)

Satisfies p(x,y,y) ≈ x, p(x,y,x) ≈ x and p(y,y,x) ≈ x. Verification is a short computation with the distributive and complementation laws.

Permutability from the Pixley term
Dropping the middle identity leaves the Mal'cev conditions, so the same term is a Mal'cev term and congruences permute.
Distributivity likewise
Con B is distributive for every Boolean algebra B, so Jónsson's lemma applies to any variety generated by Boolean algebras.
Consequence: Jónsson's lemma bites hard
Since the only subdirectly irreducible is 2, and 2 generates the whole variety, Jónsson's lemma confirms there is nothing else to find.
Consequence: the discriminator
The Pixley term here is in fact a discriminator term on 2, which is the entry point to discriminator varieties in the next stream.

04Congruences, ideals and filters

Boolean algebras admit a subobject coordinatisation of congruences, like groups and rings and unlike lattices generally.

Congruences ↔ ideals

Via the 0-class

Con B is isomorphic to the lattice of ideals of B. The congruence is recovered from the ideal I by relating a and b when their symmetric difference lies in I.

Congruences ↔ filters

Via the 1-class

Dually, Con B is isomorphic to the lattice of filters. Ideals and filters are exchanged by complementation.

The reason the coordinatisation works is the same as for groups: permutability plus constants in the type. The symmetric difference a △ b = (a ∧ b′) ∨ (a′ ∧ b) plays the role of a·b⁻¹, and it is exactly the ring subtraction under the Boolean ring correspondence developed on the next page.

05Atoms, atomicity and atomlessness

An atom is a minimal non-zero element. Boolean algebras divide sharply according to how many atoms they have.

Atomic

Every element above an atom

Every non-zero element is above some atom. Finite Boolean algebras and all power set algebras are atomic.

Atomless

No atoms at all

Every non-zero element strictly contains a smaller non-zero element. The countable atomless Boolean algebra is unique up to isomorphism.

Neither

Mixed

General Boolean algebras may have some atoms without being atomic. The atoms generate an ideal whose quotient may be atomless.

NoteThe countable atomless algebra is categorical

There is exactly one countable atomless Boolean algebra up to isomorphism — the algebra of clopen subsets of Cantor space, equivalently the free Boolean algebra on countably many generators modulo nothing. This ℵ₀-categoricity is the model-theoretic reason the theory of atomless Boolean algebras is complete and decidable.

06Finite Boolean algebras

The finite case is completely settled and is the model for everything Stone duality generalises.

Key resultStructure theorem for finite Boolean algebras

Every finite Boolean algebra is isomorphic to the power set algebra of its set of atoms. Consequently every finite Boolean algebra has cardinality 2ⁿ for some n, and two finite Boolean algebras are isomorphic exactly when they have the same number of atoms.

The proof is the subdirect representation specialised: a finite Boolean algebra embeds in a power of 2, and finiteness forces the embedding to be onto the full power set of the atom set. Stone duality is precisely the extension of this to the infinite case, where the atom set is replaced by a topological space of ultrafilters — necessary because atomless algebras have no atoms to index by.

The free Boolean algebra
Generators	Elements	Reason
n	2^2ⁿ	one element per Boolean function of n variables
countably many	countable	union of the finite free algebras

The doubly exponential free spectrum is the fastest growth possible for a locally finite variety, and it reflects that every Boolean function on n arguments is a term operation.

Frequently asked

Why is the two-element algebra so decisive?

Because it is the only subdirectly irreducible member, Birkhoff's subdirect representation theorem forces every Boolean algebra to be a subdirect power of it. No other information about the variety is needed — the single algebra determines everything, and questions about Boolean algebras reduce to questions about fields of sets.

Are complemented distributive lattices the same as Boolean algebras?

The same objects, a different type. Complemented distributive lattices in type (2, 2) with existence axioms do not form a variety; promoting complementation to a unary operation gives the Boolean algebra type and does. Since complements are unique under distributivity, no object is gained or lost — only the subalgebra and homomorphism notions change.

Is every infinite Boolean algebra a power set algebra?

No. Power set algebras are complete and atomic; the algebra of finite and cofinite subsets of an infinite set is neither. Every Boolean algebra embeds in a power set algebra, but the embedding is rarely onto, and characterising the image is exactly what Stone duality does.

Sources and further reading

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.