Definition:Bounded Below

[edit] Ordered Set

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

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

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

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

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

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

Then $f$ is said to be bounded below (in $S$ ) 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 $L \in S$ then $f$ is unbounded below (in $S$ ).