Definition:Bounded Above

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Definition

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 $f: S \to T$ be a mapping whose codomain is a poset $\left({T, \preceq}\right)$.

Then $f$ is said to be bounded above (in $T \ $) 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 $T \ $).