Definition:Bounded Above

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[edit] Ordered Set

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).


[edit] Mapping

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


Then f is said to be bounded above (in S) 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 S).

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