Definition:Max Operation/General Definition

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 $S^2$.

Real Numbers
The operation is often seen in the context of real numbers: