\begin{definition}[Definition:Pointwise Inequality]
Let $S$ be a set, and let $\struct {T, \preceq}$ be an ordered set.
Let $f, g: S \to T$ be mappings.
Then '''$f$ pointwise precedes $g$''', denoted $f \preceq g$, {{iff}}:
:$\forall s \in S: \map f s \preceq \map g s$
Thence it can be seen that '''pointwise precedence''' is an instance of an induced relation on mappings.
{{wtd|Not entirely happy with the naming}}
\end{definition}