Boolean Algebras and Stone Duality
Atoms and Finite Boolean Algebras
Atoms as the minimal non-zero elements, the classification of finite Boolean algebras as power sets, and the failure of that classification in the infinite case.
Learning objectives
- Define atom and atomic Boolean algebra
- Prove that every finite Boolean algebra is a power set
- Identify where the argument fails for infinite algebras
Atoms
An element a ≠ 0 of a Boolean algebra such that no element lies strictly between 0 and a.
One in which every non-zero element lies above some atom.
In a power set Su(X) the atoms are the singletons. Every non-empty subset contains a singleton, so power sets are atomic.
The finite classification
Every finite Boolean algebra is isomorphic to Su(X) for a finite set X — namely the set of its atoms. Consequently every finite Boolean algebra has cardinality 2n for some n.
- Atoms exist. In a finite algebra any descending chain terminates, so below any non-zero element there is a minimal non-zero one.
- Every element is a join of atoms. For non-zero b, let c be the join of the atoms below b. If c ≠ b then b ∧ c′ is non-zero and contains an atom below b but not below c, a contradiction.
- The representation is unique. Distinct sets of atoms have distinct joins, because an atom below a join of atoms must equal one of them — a consequence of distributivity.
- Conclusion. The map sending b to the set of atoms below it is an isomorphism onto the power set of the atom set.
Finite Boolean algebras are classified up to isomorphism by a single natural number, the number of atoms. There is exactly one of each size 2n and none of any other size.
Where the infinite case differs
- Atoms may not exist. Atomless Boolean algebras exist — the regular open algebra of the real line is one. Every non-zero element splits.
- Joins of atoms may not exist. Without completeness there is no guarantee that an infinite family of atoms has a join.
- Atomic does not imply power set. The finite–cofinite algebra on an infinite set is atomic, but is not a power set — the infinite coinfinite sets are missing.
| Atomic | Atomless | |
|---|---|---|
| Complete | Power sets | Regular open algebra of the reals |
| Incomplete | Finite–cofinite algebra | The free countably generated Boolean algebra |
A Boolean algebra is isomorphic to a power set if and only if it is complete and atomic.
Atomless algebras
A Boolean algebra with no atoms: every non-zero element strictly dominates another non-zero element.
Any two countable atomless Boolean algebras are isomorphic.
The countable atomless Boolean algebra is the free Boolean algebra on countably many generators, and also the Lindenbaum algebra of propositional logic with countably many variables. It is the Boolean-algebraic analogue of the rationals as the unique countable dense linear order without endpoints.
Atomless complete Boolean algebras are the standard setting for forcing in set theory. The absence of atoms is precisely what allows a generic filter to avoid deciding everything in advance.
Frequently asked questions
Can a Boolean algebra have exactly one atom?
Yes, but then it need not be small — an algebra can have a single atom together with an atomless part above it. Such algebras are neither atomic nor atomless.
Is every infinite Boolean algebra of size 2^κ for some κ?
No. The finite–cofinite algebra on a countable set is countable, and no power set is countably infinite. Cardinality constraints apply only to complete atomic algebras.
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 133-136.
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.