In mathematics, given a subset S of a totally or partially ordered set T, the supremum (sup) of S, if it exists, is the least element of T which is greater than or equal to any element of S. Consequently, the supremum is also referred to as the least upper bound (lub or LUB). If the supremum exists, it is unique. If S contains a greatest element, then that element is the supremum; otherwise, the supremum does not belong to S (or does not exist). For instance, the negative real numbers do not have a greatest element, and their supremum is 0 (which is not a negative real number).

Suprema are often considered for subsets of real numbers, rational numbers, or any other well-known mathematical structure for which it is immediately clear what it means for an element to be "greater-than-or-equal-to" another element. The definition generalizes easily to the more abstract setting of order theory, where one considers arbitrary partially ordered sets.

The concept of supremum coincides with the concept of least upper bound, but not with the concepts of minimal upper bound, maximal element, or greatest element. The supremum is in a precise sense dual to the concept of an infimum.

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