Definition:Strictly Monotone/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 strictly monotone if and only if it is either strictly increasing or strictly decreasing.

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

Also known as

This can also be called strictly monotonic.

Also see

• Definition:Increasing Mapping
• Definition:Decreasing Mapping
• Definition:Monotone Mapping

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): $\S 14$