# Definition:Supremum

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## Disambiguation

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**Supremum** may refer to:

## 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 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 $\bigvee T$ or $\ds \bigvee_{y \mathop \in T} y$.

Some sources introduce the notation $\ds \sup_{y \mathop \in T} y$, 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 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.

## Sources

- 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**least upper bound (l.u.b.; supremum)** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**least upper bound (l.u.b.; supremum)**