The supremum of a set $X$ having a partial order is the least upper bound of $X$ (if it exists) and is denoted $\sup{X}$

Let $A$ be a set with a partial order $\leq$ and let $X \subseteq A$ Then $s = \sup X$ if and only if:

1.

For all $x\in X$ we have $x \leq s$ (i.e. $s$ is an upper bound).

2.

If $s^{\prime}$ meets condition 1, then $s \leq s^{\prime}$ ($s$ is the least upper bound).

There is another useful definition which works if $A = \mathbb{R}$ with $\leq$ the usual order on $\mathbb{R}$ supposing that s is an upper bound: $$s = \sup X \text{ if and only if } \forall \varepsilon > 0, \exists x\in X : s-\varepsilon < x.$$

Note that it is not necessarily the case that $\sup X \in X$ Suppose $X = {]0, 1[}$ then $\sup X = 1$ but $1 \not\in X$

Note also that a set may not have an upper bound at all.