Definition:Max Operation

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

The max operation is the binary operation on $\left({S, \preceq}\right)$ defined as:
 * $\forall x, y \in S: \max \left({x, y}\right) = \begin{cases}

y & : x \preceq y \\ x & : y \preceq 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.