Definition:Min Operation

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

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

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