\begin{definition}[Definition:Monotonicity]
Let $\left({S, \preceq_1}\right)$ and $\left({T, \preceq_2}\right)$ be ordered sets.
Let $\phi: \left({S, \preceq_1}\right) \to \left({T, \preceq_2}\right)$ be a monotone mapping.
Then whether $\phi$ is increasing or decreasing is known as the '''monotonicity of $\phi$'''. 
That is:
:if $\phi$ is such that $x \mathop{\preceq_1} y \implies \phi \left({x}\right) \mathop{\preceq_2} \phi \left({y}\right)$, then the '''monotonicity''' of $\phi$ is increasing
:if $\phi$ is such that $x \mathop{\prec_1} y \implies \phi \left({x}\right) \mathop{\prec_2} \phi \left({y}\right)$, then the '''monotonicity''' of $\phi$ is strictly increasing
:if $\phi$ is such that $x \mathop{\preceq_1} y \implies \phi \left({y}\right) \mathop{\preceq_2} \phi \left({x}\right)$, then the '''monotonicity''' of $\phi$ is decreasing
:if $\phi$ is such that $x \mathop{\prec_1} y \implies \phi \left({y}\right) \mathop{\prec_2} \phi \left({x}\right)$, then the '''monotonicity''' of $\phi$ is strictly decreasing.
\end{definition}