# Definition:Supremum of Set

This page is about Supremum in the context of Ordered Set. For other uses, see Supremum.

## Definition

Let $\struct {S, \preccurlyeq}$ be an ordered set.

Let $T \subseteq S$.

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

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

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

$T$ admits a supremum (in $S$) or
$T$ has a supremum (in $S$).

### Finite Supremum

If $T$ is finite, $\sup T$ is called a finite supremum.

### Subset of Real Numbers

The concept is usually encountered where $\struct {S, \preccurlyeq}$ is the set of real numbers under the usual ordering $\struct {\R, \le}$:

Let $T \subseteq \R$ be a subset of the real numbers.

A real number $c \in \R$ is the supremum of $T$ in $\R$ if and only if:

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

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

## Also known as

Particularly in the field of analysis, the supremum of a set $T$ is often referred to as the least upper bound of $T$ and denoted $\map {\mathrm {lub} } T$ or $\map {\mathrm {l.u.b.} } T$.

Some sources refer to the supremum of a set as the supremum on a set.

Some sources refer to the supremum of a set as the join of the set and use the notation $\bigvee S$.

Some sources introduce the notation $\ds \sup_{y \mathop \in S} y$, which may improve clarity in some circumstances.

Some older sources, applying the concept to a (strictly) increasing real sequence, refer to a supremum as an upper limit.

## Also defined as

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.

## Also see

• Results about suprema can be found here.

## Linguistic Note

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