Definition:Monotone (Order Theory)/Mapping

Definition

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 mapping.

Then $\phi$ is monotone if and only if it is either increasing or decreasing.

Note that this definition also holds if $S = T$.

Some authors take monotone mapping to mean what on $\mathsf{Pr} \infty \mathsf{fWiki}$ is called an increasing mapping.