Definition:Bounded Below

General Poset
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) such that precedes all the elements in $$T$$.

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

Mappings
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 $$S$$.