Least upper bound

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Given a subset $S$ in some larger ordered set $R$, a least upper bound or supremum for $S$ is an element $M \in R$ such that $s \leq M$ for every $s \in S$ and there is no $m < M$ with this same property.

If the least upper bound $M$ of $S$ is an element of $S$, it is also the maximum of $S$. If $M \not\in S$, then $S$ has no maximum.