Definition:Max Operation

Let $$\left({S, \le}\right)$$ be a totally ordered set.

The max operation is the binary operation on $$\left({S, \le}\right)$$ defined as:

$$ \forall x, y \in S: \max \left({x, y}\right) = \begin{cases} y: & x \le y \\ x: & y \le x \end{cases} $$

Notation
The notation $$\max \left\{{x, y}\right\}$$ is frequently seen for $$\max \left({x, y}\right)$$.

This emphasises that the operands of the max operation are undifferentiated as to order.