Definition:Monotone (Order Theory)/Mapping

Definition
Let $\struct {S, \preceq_1}$ and $\struct {T, \preceq_2}$ be ordered sets.

Let $\phi: \struct {S, \preceq_1} \to \struct {T, \preceq_2}$ be a mapping.

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

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

Also defined as
Some authors take monotone mapping to mean what on is called an increasing mapping.

Also known as
Some sources use the term monotonic.

Also see

 * Definition:Strictly Monotone Mapping