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 $\min(A)$ .