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 if and only if 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 $\mathsf{Pr} \infty \mathsf{fWiki}$ is called an increasing mapping.

Also known as

Some sources use the term monotonic.

Also see

• Definition:Strictly Monotone Mapping

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {III}$: The Natural Numbers: $\S 14$: Orderings
• 2010: Steve Awodey: Category Theory (2nd ed.) ... (previous) ... (next): $\S 1.4.3$