Zorn's Lemma

A statement in order theory equivalent (over ZF) to the axiom of choice: in a partially ordered set in which every totally ordered subset (chain) has an upper bound, there exists at least one maximal element.

A principle in set theory that asserts: if a partially ordered set has the property that every totally ordered subset (chain) has an upper bound in the set, then the poset contains at least one maximal element (an element not strictly below any other).

A principle in set theory and order theory asserting that if a partially ordered set has the property that every totally ordered subset (chain) has an upper bound in the set, then the poset contains at least one maximal element (an element with no strictly larger element).

An axiomatically equivalent principle to the Axiom of Choice: if a partially ordered set has the property that every totally ordered subset (chain) has an upper bound, then the poset contains at least one maximal element.