\begin{definition}[Definition:Ordering on Mappings]
Let $S$ be a set.
Let $\left({T, \preceq}\right)$ be an ordered set.
Let $f, g: S \to T$ be mappings.
Then '''ordering on mappings''' $f$ and $g$ denoted $f \preceq g$ is defined by
:$\forall s \in S: f\left({s}\right) \preceq g\left({s}\right)$
\end{definition}