Let $P = (X, {\preceq})$ be an D1707: Ordered set.
The minimum operation on $X$ with respect to $P$ is the D18: Map
\begin{equation}
X \times X \to X, \quad
(x, y) \mapsto
\begin{cases}
x, \quad & (x, y) \in {\preceq} \\
y, \quad & (y, x) \in {\preceq}
\end{cases}
\end{equation}