Zorn's lemma

Zorn's lemma states that a partially ordered set has a maximal element if every chain has an upper bound in that set. It is equivalent to the axiom of choice and the well-ordering theorem.

Partially ordered set: A set equipped with a relation that is reflexive, antisymmetric, and transitive. Zorn's lemma begins with this structure and asks whether it contains a maximal element.
Hamel basis: A basis of a vector space in which every vector is a unique finite linear combination of basis vectors. Zorn's lemma proves that every vector space has a Hamel basis.
Max Zorn: Max Zorn was a German-born mathematician who formulated the maximality principle now bearing his name. His 1935 paper presented the principle that became known as Zorn's lemma.
Greatest element: An element of an ordered set that is greater than or equal to every element of the set. Zorn guarantees only a maximal element, which need not dominate incomparable elements.
Chain (order theory): A subset of a partially ordered set in which every pair of elements is comparable. The hypothesis requires every such totally ordered subset to have an upper bound.
Maximal ideal: A proper ideal contained in no larger proper ideal of a ring. Applying Zorn's lemma to proper ideals proves that every nonzero unital commutative ring has a maximal ideal.
Hausdorff maximal principle: The principle that every partially ordered set has a maximal chain. Hausdorff's earlier maximality principle is closely related and equivalent in standard set theory.
Minimal element: An element of a partially ordered set with no strictly smaller element in that set. Reversing the order turns Zorn's maximal-element conclusion into a minimal-element conclusion.
Upper bound: An element that is greater than or equal to every element of a specified subset under an order. Each chain must have an upper bound in the partially ordered set.
Ultrafilter: A maximal proper filter on a set or, more generally, on a Boolean algebra. Zorn's lemma extends suitable proper filters to ultrafilters.

Linked from 12 pages