Upper bound

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Needs fleshing out, a few references and a terminology section would be good

Definition

Let $(X,\preceq)$ be a poset and let $A\in\mathcal{P}(X)$ be any subset of $X$. An element $b\in X$ is an upper bound of $A$ if:

• $\forall a\in A[a\preceq b]$.

Equivalently, a subset $A\in\mathcal{P}(X)$ has a upper bound if:

• $\exists b\in X\forall a\in A[a\preceq b]$ - "if there exists a upper bound."

Terminology

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TODO: Things like "bounded above" and such

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See also

• Supremum - the lowest upper bound of a set.
• Lower bound - the dual concept.
  • Infimum - the greatest lower bound.