Maximum of Two Elements of a Totally Ordered Set

Let $S$ be a set equipped with a total order $\le$, and let $a,b\in S$.

The maximum of $a$ and $b$, written $\max\{a,b\}$, is the element of $S$ defined as follows: if $a\le b$, then $\max\{a,b\}$ is $b$; otherwise $\max\{a,b\}$ is $a$.