\begin{definition}[Definition:Max Operation]
Let $\struct {S, \preceq}$ be a totally ordered set.
The '''max operation''' is the binary operation on $\struct {S, \preceq}$ defined as:
:$\forall x, y \in S: \map \max {x, y} = \begin{cases}
y & : x \preceq y \\
x & : y \preceq x
\end{cases}$
\end{definition}