# 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 $\min(A)$.

Title lowest upper bound LowestUpperBound 2013-03-22 11:52:18 2013-03-22 11:52:18 djao (24) djao (24) 13 djao (24) Definition msc 06A05 least upper bound greatest lower bound supremum infimum