\begin{definition}[Definition:Min Operation]
Let $\struct {S, \preceq}$ be a totally ordered set.
The '''min operation''' is the binary operation on $\struct {S, \preceq}$ defined as:
:$\forall x, y \in S: \map \min {x, y} = \begin {cases}
x & : x \preceq y \\
y & : y \preceq x
\end {cases}$
\end{definition}