\begin{definition}[Definition:Increasing/Mapping]
Let $\struct {S, \preceq_1}$ and $\struct {T, \preceq_2}$ be ordered sets.
Let $\phi: S \to T$ be a mapping.
Then $\phi$ is '''increasing''' {{iff}}:
:$\forall x, y \in S: x \preceq_1 y \implies \map \phi x \preceq_2 \map \phi y$
Note that this definition also holds if $S = T$.
\end{definition}