Stone's representation theorem for finite Boolean algebras

Statement

An atom of a Boolean algebra $B$ is a minimal element of $B \setminus \{0\}$. Let $B$ be a finite Boolean algebra, and let $\operatorname{At}(B)$ be its set of atoms. Then the map $b \mapsto \{ x \in \operatorname{At}(B) : x \le b \}$ is an isomorphism from $B$ to the power set Boolean algebra of $\operatorname{At}(B)$.

Remarks

A consequence: a finite Boolean algebra with $n$ atoms has $2^n$ elements. The general theorem says that every Boolean algebra is isomorphic to the algebra of clopen subsets of its Stone space. For a finite Boolean algebra this space is finite and discrete, and its points correspond to the atoms. The general theorem needs a weak form of the axiom of choice, but the finite case does not.

History

Marshall Stone published the representation theorem for all Boolean algebras in 1936.

Explanations

L1 human

Each finite Boolean algebra is the algebra of all subsets of some finite set. Each element is the set of atoms below it, and the atoms are the smallest nonzero elements.