Boolean prime ideal theorem

Every nontrivial Boolean algebra has a prime ideal. Equivalent to the ultrafilter lemma, it is weaker than the axiom of choice.

Ultrafilter lemma: Every proper filter on a set extends to an ultrafilter. This set-theoretic principle is equivalent to the Boolean prime ideal theorem.
Boolean algebra: An algebraic structure with meet, join, and complement operations satisfying the laws of classical logic. The theorem applies to every nontrivial structure of this kind.
Boolean ring: A commutative ring in which every element is idempotent under multiplication. Boolean algebras can be represented as Boolean rings, translating the theorem into ring language.
Stone duality: A dual equivalence between Boolean algebras and compact Hausdorff totally disconnected spaces. Prime ideals or ultrafilters provide the points used to represent Boolean algebras as spaces.
Boolean prime ideal theorem for distributive lattices: The principle that every proper ideal disjoint from a filter in a distributive lattice extends to a prime ideal. This distributive-lattice version is a broader formulation equivalent to the Boolean theorem.
Axiom of choice: Every family of nonempty sets has a choice function. The theorem follows from choice, but choice is strictly stronger.
Filter (mathematics): A nonempty upward-closed subset of a partially ordered set that is closed under finite meets. The proof starts with a proper filter and extends it maximally.
Lattice (order): A partially ordered set in which every pair has a least upper bound and a greatest lower bound. Boolean algebras are complemented distributive lattices, the setting behind their ideals.
Stone space: The compact Hausdorff space of ultrafilters on a Boolean algebra, with basic clopen sets determined by its elements. The theorem ensures the ultrafilter space has points whenever the algebra is nontrivial.
Hausdorff maximal principle: Every partially ordered set has a maximal chain. This maximality principle is equivalent to the axiom of choice, unlike the weaker theorem here.

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