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Homelowest upper bound

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# lowest upper bound

Let $S$ be a set with a partial ordering $\leq$, and let $T$ be a subset of $S$. A *lowest upper bound*, or *supremum*, of $T$ is an upper bound $x$ of $T$ with the property that $x\leq y$ for every upper bound $y$ of $T$. The lowest upper bound of $T$, when it exists, is denoted $\operatorname{sup}(T)$.

A lowest upper bound of $T$, when it exists, is unique.

Greatest lower bound is defined similarly: a *greatest lower bound*, or *infimum*, of $T$ is a lower bound $x$ of $T$ with the property that $x\geq y$ for every lower bound $y$ of $T$. The greatest lower bound of $T$, when it exists, is denoted $\operatorname{inf}(T)$.

If $A=\{a_{1},a_{2},\ldots,a_{n}\}$ is a finite set, then the supremum of $A$ is simply $\max(A)$, and the infimum of $A$ is equal to $\min(A)$.

## Mathematics Subject Classification

06A05*no label found*

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