Definition:Supremum of Mapping

Definition
Let $S$ be a set.

Let $\struct {T, \preceq}$ be an ordered set.

Let $f: S \to T$ be a mapping from $S$ to $T$.

Let $f \sqbrk S$, the image of $f$, admit a supremum.

Then the supremum of $f$ (on $S$) is defined by:
 * $\displaystyle \sup_{x \mathop \in S} \map f x = \sup f \sqbrk S$

Also see

 * Definition:Infimum of Mapping