Maximum of a subset of a total order ($\max S$)

A maximum of a subset $S$ of a total order is an element $m\in S$ with $s\le m$ for every $s\in S$. If it exists it is unique. Every nonempty finite subset has a maximum, by repeated pairwise comparison. Unlike a supremum, a maximum must belong to the subset. The completion time of jobs processed on finitely many parallel machines is the maximum of the machine completion times.