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 $\leqslant$, and let $X\subseteq A$. Then $s=\sup X$ if and only if:

1. 1.

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

2. 2.

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

There is another useful definition which works if $A=\mathbb{R}$ with $\leqslant$ the usual order on $\mathbb{R}$, supposing that s is an upper bound:

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.