Definition:Bounded Below/Mapping

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Definition

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

Then $f$ is said to be bounded below (in $T \ $) by the lower bound $L$ iff:

$\forall x \in S: L \preceq f \left({x}\right)$

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

If there is no such $L \in T$ then $f$ is unbounded below (in $T \ $).