Definition:Bounded Above

[edit] Ordered Set

Let $\left({S; \preceq}\right)$ be a poset.

A subset $T \subseteq S$ is bounded above (in $S$ ) if:

$\exists M \in S: \forall a \in T: a \preceq M$

That is, there is an element of $S$ (at least one) that succeeds all the elements in $T$ .

If there is no such element, then $T$ is unbounded above (in $S$ ).

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 $S$ ).