Maximal element

A maximal element of a partially ordered set has no strictly greater element in that set. It need not be greater than every other element.

Partially ordered set: A set equipped with a reflexive, antisymmetric, transitive relation. Maximality is defined relative to the order relation on this set.
Minimal element: An element of a partially ordered set with no strictly smaller element in that set. It reverses the direction of the defining comparison.
Maximal ideal: A proper ideal of a ring that is contained in no larger proper ideal. It is maximality in the partially ordered set of proper ideals ordered by inclusion.
Hasse diagram: A diagram representing a finite partially ordered set by drawing only its cover relations. Maximal elements appear as vertices with no upward path to a larger element.
Axiom of choice: The principle that a family of nonempty sets has a choice function selecting one member from each set. Zorn's lemma, a standard maximal-element principle, is equivalent to the axiom of choice.
Partial order: A binary relation that is reflexive, antisymmetric, and transitive. Its comparisons decide whether any element lies strictly above a candidate.
Least element: An element less than or equal to every element of an ordered set. A least element is globally below all elements, stronger than being minimal.
Maximal matching: A matching in a graph to which no edge can be added while preserving the matching property. Matchings ordered by inclusion make maximal matchings concrete examples of the definition.
Cover relation: A relation where one ordered element lies immediately below another, with no element between them. Cover edges reveal which elements can be reached by upward steps.
Hausdorff maximal principle: The principle that every partially ordered set has a maximal chain. It guarantees maximal chains, which are maximal objects in an inclusion order.

Linked from 8 pages