SUPREMUM

Etymology

Noun

supremum (plural suprema)

(set theory) (real analysis): Given a subset X of R, the smallest real number that is ≥ every element of X; (order theory): given a subset X of a partially ordered set P (with partial order ≤), the least element y of P such that every element of X is ≤ y.

Usage notes

• Commonly denoted sup(X).

• The supremum of X may not exist, and, if it does, may not be an element of X.

• (order theory)

Formally: Let $$S =\{ t : t \in P : \forall x\in X, x \le t \}$$ be the set of upper bounds of X. Then sup(X), if it exists, is the element $$s\in S: \forall y\in S, s \le y$$ .

The concept of supremum is closely related to the function ∨ (called join). The supremum of two elements, denoted $$\sup\{x,y\}$$ can also be written $$x\lor y$$ . The supremum of a set may be denoted $$\sup(X)$$ or $$\bigvee X$$ .