Order, lattices, ordered algebraic structures Theorem

Stone's representation theorem for finite Boolean algebras

Builds on Boolean algebra Power set Boolean algebra

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.

Reading path

To understand this page from the ground up, read these 3 pages in order. The foundations come first.

Show the path as a graph

Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

no evidence level

Boolean algebra definition
Power set Boolean algebra example
Stone's representation theorem for finite Boolean algebras statement

Tag	`0DYS9C3HH5`
URI	`m2://0DYS9C3HH5`
Kind	statement, theorem
Writing	W0 stub
MSC	06E05, 06E15
Status	This copy of the site has no build data. Run `m2 status 0DYS9C3HH5`.
Uses	Boolean algebra, Power set Boolean algebra

Stone's representation theorem for finite Boolean algebras

Statement

Remarks

History

Reading path

Further reading

What this page needs for W1

What links here