\begin{definition}[Definition:Max Operation/General Definition]
Let $\struct {S, \preceq}$ be a totally ordered set.
Let $S^n$ be the cartesian $n$th power of $S$.
The '''max operation''' is the $n$-ary operation on $\struct {S, \preceq}$ defined recursively as:
:$\forall x := \family {x_i}_{1 \mathop \le i \mathop \le n} \in S^n: \map \max x = \begin{cases}
x_1 & : n = 1 \\
\map \max {x_1, x_2} & : n = 2 \\
\map \max {\map \max {x_1, \ldots, x_{n - 1} }, x_n} & : n > 2 \\
\end{cases}$
where $\map \max {x, y}$ is the binary max operation on $S^2$.
\end{definition}