# Definition:Max Operation/General Definition

## Definition

Let $\left({S, \preceq}\right)$ 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 $\left({S, \preceq}\right)$ defined recursively as:

$\forall x := \left \langle{x_i}\right \rangle_{1 \mathop \le i \mathop \le n} \in S^n: \max \left({x}\right) = \begin{cases} x_1 & : n = 1 \\ \max \left({x_1, x_2}\right) & : n = 2 \\ \max \left({\max \left({x_1, \ldots, x_{n-1}}\right), x_n}\right) & : n > 2 \\ \end{cases}$

where $\max \left({x, y}\right)$ is the binary max operation on $S^2$.

### Real Numbers

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

The max operation on the real cartesian space $\R^n$ is the real-valued function $\max: \R^n \to \R$ defined recursively as:

$\forall x := \left \langle{x_i}\right \rangle_{1 \mathop \le i \mathop \le n} \in \R^n: \max \left({x}\right) = \begin{cases} x_1 & : n = 1 \\ \max \left({x_1, x_2}\right) & : n = 2 \\ \max \left({\max \left({x_1, \ldots, x_{n-1}}\right), x_n}\right) & : n > 2 \\ \end{cases}$

where $\max \left({x, y}\right)$ is the binary max operation on $\R \times \R$.