Definition:Supremum

[edit] Ordered Set

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:

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 $\operatorname{lub} \left({T}\right)$ .

Let $f$ be a mapping defined on a poset $\left({S, \preceq}\right)$ .

Let $f$ be bounded above on $S$ .

It follows from the Continuum Property that the codomain of $f$ has a supremum on $S$ .

Thus:

$\sup_{x \in S} f \left({x}\right) = \sup f \left({S}\right)$ .