supremums

Supremums, or suprema, are a fundamental concept in order theory and analysis. For a subset S of a partially ordered set, a supremum (least upper bound) is an element u that satisfies two conditions: every element of S is less than or equal to u (u is an upper bound of S), and no smaller element than u is an upper bound of S. If such a u exists, it is unique.

In the real numbers, the supremum of a nonempty set that is bounded above always exists; this is the completeness property of the real numbers. If the supremum belongs to the set, the supremum is also the maximum of S. If not, the supremum is not contained in S. For example, for S = (0,1) the supremum is 1, but S has no maximum. For S = {x ∈ R : x ≤ 3}, the supremum is 3 and the maximum is 3.

The concept is dual to the infimum, or greatest lower bound, and it extends to broader ordered structures such as lattices. A set is bounded above when a supremum exists, and in certain lattices the least upper bound property characterizes completeness.

Key properties include: if S ⊆ T and sup T exists, then sup S ≤ sup T; the supremum of a finite nonempty set is its maximum. Supremums are central in definitions of limits, optimization, and integration, and are used in various branches of analysis to describe limiting and bounding behavior. Notation commonly uses sup S, with suprema or supremums as plural forms.