Suprema

Supremum is a concept from order theory describing the least upper bound of a set within a partially ordered context. Let P be a set equipped with a reflexive, antisymmetric, transitive relation ≤, and let A be a subset of P. An element u in P is an upper bound of A if a ≤ u for every a in A. The supremum (plural: suprema) of A, when it exists, is the smallest such upper bound: u is a supremum of A if it is an upper bound and, for any other upper bound v of A, we have u ≤ v. If the supremum belongs to A itself, A is said to have a maximum, and that maximum equals the supremum.

Suprema are dual to infima. An infimum (greatest lower bound) of a subset B of P is the largest element that is not less than any element of B; it is the infimum if and only if it exists. A set may have a supremum without having a maximum, and some sets may lack a supremum altogether in a given poset.

In the real numbers with the usual order, every nonempty subset that is bounded above has a supremum. This is the least upper bound property, which makes the real numbers a complete ordered field. Examples include sup {x ∈ R : x^2 < 2} = √2 and sup (-∞, 1) = 1, where the supremum may be attained by the bound or only approached by elements of the set.

More generally, whether a supremum exists depends on the ambient poset. Complete lattices guarantee the existence of suprema for all subsets, while in many other ordered sets suprema may fail to exist.