Greatest element and least element

In an ordered set, a greatest element is greater than or equal to every element, and a least element is less than or equal to every element. Either may be absent, but if present it is unique.

Partially ordered set: A set equipped with a reflexive, antisymmetric, transitive relation. Greatest and least elements are defined relative to this broader kind of ordered structure.
Hasse diagram: A diagram of a finite partially ordered set showing its cover relations. A greatest element appears above every other vertex, and a least element below every other vertex.
Natural number: A member of the number system used for counting, with conventions about whether zero is included. Under the usual order, the natural numbers have a least element but no greatest element.
Supremum: The least upper bound of a set in an ordered structure, whether or not it belongs to the set. A supremum may exist without being a greatest element because it can lie outside the set.
Total order: A partial order in which every pair of elements is comparable. In a total order, greatest and least elements coincide with the familiar largest and smallest values.
Lattice (order theory): A partially ordered set in which every pair has a least upper bound and a greatest lower bound. Lattices provide pairwise join and meet operations that generalize greatest and least elements.
Closed interval: A real interval that includes both of its finite endpoints. Its endpoints are respectively the least and greatest elements under the usual numerical order.
Infimum: The greatest lower bound of a set in an ordered structure, whether or not it belongs to the set. An infimum may exist without being a least element because it can lie outside the set.
Upper bound: An element greater than or equal to every member of a specified subset. A greatest element is an upper bound for the entire ordered set and belongs to it.
Top element: An element greater than or equal to every element in an ordered structure. This is the order-theoretic name commonly used for a greatest element.

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Partial orderRelated: Unlike minimal or maximal elements, these are unique when they exist.