Boolean Algebra and Lattices - Main Theorem

Stone's Representation Theorem reveals the deep connection between abstract Boolean algebras and set algebras, showing every Boolean algebra is essentially a field of sets.

DefinitionStone's Representation Theorem

Every Boolean algebra is isomorphic to a field of sets (a Boolean algebra of subsets of some set with operations $\cap, \cup, \overline{\phantom{A}}$ ).

More precisely, every Boolean algebra $B$ is isomorphic to a subalgebra of $\mathcal{P}(X)$ for some set $X$ .

Proof Sketch:

The key is to construct a suitable set $X$ and embedding $\phi: B \to \mathcal{P}(X)$ .

Step 1: Define $X$ .

Let $X$ be the set of all ultrafilters on $B$ . An ultrafilter $U$ is a subset of $B$ satisfying:

$1 \in U$ and $0 \notin U$
If $a, b \in U$ , then $a \land b \in U$ (closed under meet)
If $a \in U$ and $a \leq b$ , then $b \in U$ (upward closed)
For every $a \in B$ , either $a \in U$ or $\neg a \in U$ (but not both)

Step 2: Define the embedding.

For each $a \in B$ , define $\phi(a) = \{U \in X : a \in U\}$ (the set of ultrafilters containing $a$ ).

Step 3: Verify $\phi$ is a homomorphism.

$\phi(a \land b) = \{U : a \land b \in U\} = \{U : a \in U \text{ and } b \in U\} = \phi(a) \cap \phi(b)$
$\phi(a \lor b) = \phi(a) \cup \phi(b)$ (similarly)
$\phi(\neg a) = \{U : \neg a \in U\} = \{U : a \notin U\} = X \setminus \phi(a) = \overline{\phi(a)}$

Step 4: Verify $\phi$ is injective.

If $\phi(a) = \phi(b)$ , then every ultrafilter contains $a$ iff it contains $b$ . This implies $a = b$ (by properties of Boolean algebras and ultrafilters).

Therefore, $B$ is isomorphic to its image $\phi(B) \subseteq \mathcal{P}(X)$ . $\square$

ExampleFinite Boolean Algebras

For finite Boolean algebras, the theorem is simpler. A finite Boolean algebra with $n$ atoms (minimal non-zero elements) is isomorphic to $\mathcal{P}(A)$ where $A$ is the set of atoms, and $|B| = 2^n$ .

Every element is uniquely a join of atoms: if $a = a_1 \lor a_2 \lor \cdots \lor a_k$ (atoms), then $\phi(a) = \{a_1, a_2, \ldots, a_k\}$ .

Remark

Stone's theorem generalizes to Stone duality, establishing a correspondence between Boolean algebras and certain topological spaces (Stone spaces: compact, Hausdorff, totally disconnected). This connects algebra with topology and has applications in logic (completeness theorems) and theoretical computer science (domain theory). The theorem shows that despite the abstract axiomatic definition, Boolean algebras fundamentally capture the structure of set operations. In finite cases, this means every Boolean algebra is a power set algebra, explaining why power sets are the canonical example.