Boolean Algebras and Stone Duality
Boolean Algebras: Axioms and First Examples
Boolean algebras as a variety: the axioms, the two-element algebra that generates everything, and the examples that motivate the theory.
Learning objectives
- State the Boolean algebra axioms
- Verify the standard examples
- Explain why the two-element algebra is the whole story
The axioms
An algebra ⟨B, ∨, ∧, ′, 0, 1⟩ of type ⟨2, 2, 1, 0, 0⟩ satisfying: the lattice axioms L1–L4; distributivity; the bound laws x ∨ 0 ≈ x and x ∧ 1 ≈ x; and the complement laws x ∨ x′ ≈ 1 and x ∧ x′ ≈ 0.
Every axiom is an identity, so by Birkhoff's theorem the class is closed under H, S and P, free Boolean algebras exist, and the whole apparatus of Chapter II applies. Complementation is included as an operation precisely so that subalgebras are closed under it.
First examples
| Algebra | Operations | Notes |
|---|---|---|
| 2 = {0, 1} | Truth tables | The two-element algebra; generates the variety |
| Su(X) — the power set | ∪, ∩, complement | The motivating example; every finite Boolean algebra is of this form |
| Clopen subsets of a topological space | ∪, ∩, complement | The dual side of Stone duality |
| Finite and cofinite subsets of an infinite set | ∪, ∩, complement | An infinite Boolean algebra that is not a power set |
| Regular open sets of a topological space | Modified operations | Complete, and important in forcing |
| Lindenbaum algebra of a propositional theory | Induced by ∨, ∧, ¬ | Formulas modulo provable equivalence |
It shows that infinite Boolean algebras need not be power sets. The power-set algebras are exactly the complete atomic Boolean algebras, and the finite–cofinite algebra is atomic but not complete.
Basic consequences
In a Boolean algebra, complements are unique: if x ∨ y = 1 and x ∧ y = 0, then y = x′.
The proof uses distributivity: y = y ∧ 1 = y ∧ (x ∨ x′) = (y ∧ x) ∨ (y ∧ x′) = 0 ∨ (y ∧ x′), and a symmetric computation gives the reverse inequality.
A lattice can be bounded and complemented without being Boolean — M5 is complemented but not distributive, and its complements are not unique. Distributivity is what forces uniqueness, which is why complementation can be an operation at all.
Why 2 generates everything
Every Boolean algebra lies in HSP({2}). Indeed 2 is the only subdirectly irreducible Boolean algebra, so every Boolean algebra is a subdirect power of 2.
Because identities need only be checked in 2, verifying a Boolean identity in n variables is a check of 2n rows — a truth table. This is the algebraic reason truth tables work.
Frequently asked questions
Is every Boolean algebra a power set?
No. Only the complete atomic ones are. The finite–cofinite algebra on an infinite set, and the Lindenbaum algebra of a first-order theory, are counterexamples. Every Boolean algebra does embed in a power set, which is the Stone representation theorem.
Why include 0 and 1 as nullary operations?
So that subalgebras contain them. A sublattice of a Boolean algebra need not contain the bounds, and would then fail to be a Boolean algebra.
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.1, book pages 129-131.
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.