Definition:Supremum of Set

Definition
Let $\left({S, \preceq}\right)$ be a poset.

Let $T \subseteq S$.

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


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

Plural: Suprema.

The supremum of $T$ is denoted $\sup \left({T}\right)$.

The supremum of $x_1, x_2, \ldots, x_n$ is denoted $\sup \left\{{x_1, x_2, \ldots, x_n}\right\}$.

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

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

Also see

 * Infimum

Variants of Definition
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.