Definition:Bounded Below

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[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).


[edit] Mapping

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).

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