Definition:Supremum of Set

Definition
Let $\left({S, \preceq}\right)$ be an ordered set.

Let $T \subseteq S$.

An element $c \in S$ is the supremum of $T$ in $S$ iff:


 * $(1): \quad c$ is an upper bound of $T$ in $S$
 * $(2): \quad c \preceq d$ for all upper bounds $d$ of $T$ in $S$.

The supremum of $T$ is denoted $\sup T$.

If there exists a supremum of $T$ (in $S$), we say that:
 * $T$ admits a supremum (in $S$) or
 * $T$ has a supremum (in $S$).

Subset of Real Numbers
The concept is usually encountered where $\left({S, \preceq}\right)$ is the set of real numbers under the usual ordering: $\left({\R, \le}\right)$:

Also known as
The supremum of $T$ is often called the least upper bound of $T$ and denoted $\operatorname{lub} \left({T}\right)$.

Some sources refer to the supremum as being the upper bound. Using this convention, any element greater than this is not considered to be an upper bound.

Linguistic Note
The plural of supremum is suprema, although the (incorrect) form supremums can occasionally be found if you look hard enough.

Also see

 * Definition:Infimum of Set


 * Supremum and Infimum are Unique