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
Canonical name	LowestUpperBound
Date of creation	2013-03-22 11:52:18
Last modified on	2013-03-22 11:52:18
Owner	djao (24)
Last modified by	djao (24)
Numerical id	13
Author	djao (24)
Entry type	Definition
Classification	msc 06A05
Defines	least upper bound
Defines	greatest lower bound
Defines	supremum
Defines	infimum