Definition:Strictly Monotone

Let $$\left({S; \le_1}\right)$$ and $$\left({T; \le_2}\right)$$ be posets.

Let $$\phi: \left({S; \le_1}\right) \to \left({T; \le_2}\right)$$ be a mapping.

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

This can also be called "strictly monotonic".