supremal

Supremal is an adjective relating to the supremum, the least upper bound of a set in a partially ordered set, typically the real numbers. In real analysis, for a nonempty set S that is bounded above, its supremum sup S is the smallest real number that is an upper bound for S. If S contains a largest element, that element is both the maximum and the supremum; otherwise the supremum may not belong to S. The concept relies on the completeness of the real numbers.

In common usage, supremal describes properties or quantities associated with the supremum. For example, one might refer to the supremal value of a function on a domain or to a supremal bound. The term is often used interchangeably with supremum in contexts where an attribute or condition is defined in terms of the least upper bound.

Differences with related concepts are important: the supremum may exist without the set containing it (as in S = (0,1), where sup S = 1 but 1 ∉ S), and a set may be unbounded above, in which case its supremum is not finite (in extended real terms, it is +∞). The dual notion is the infimum, the greatest lower bound, and together they form key ideas in order theory and analysis.

Supremal is less common in everyday prose than supremum, but it appears in mathematical writing to emphasize connection with the least upper bound or extremal properties.