Definition:Bounded Above Mapping

Definition
Let $f$ be a mapping defined on a poset $\left({S, \preceq}\right)$.

Then $f$ is said to be bounded above (in $S$) by the upper bound $H$ iff:
 * $\forall x \in S: f \left({x}\right) \preceq H$.

That is, iff $f \left({S}\right) = \left\{{f \left({x}\right): x \in S}\right\}$ is bounded above by $H$.

If there is no such $H \in S$ then $f$ is unbounded above (in $S$).

Also see

 * Bounded Below
 * Bounded