Definition:Max Operation/General Definition

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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:


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

$\forall x := \family {x_i}_{1 \mathop \le i \mathop \le n} \in \R^n: \map \max x = \begin{cases} x_1 & : n = 1 \\ \map \max {x_1, x_2} & : n = 2 \\ \map \max {\map \max {x_1, \ldots, x_{n - 1} }, x_n} & : n > 2 \\ \end{cases}$

where $\map \max {x, y}$ is the binary max operation on $\R \times \R$.