Mapping on Integers is Endomorphism of Max or Min Operation iff Increasing

Theorem
Let $\Z$ denote the set of integers.

Let $f: \Z \to \Z$ be a mapping on $\Z$.

Let $\vee$ and $\wedge$ be the operations on $\Z$ defined as:

Then:
 * $f$ is an endomorphism of $\struct {\Z, \vee}$ or $\struct {\Z, \wedge}$


 * $f$ is an increasing mapping.
 * $f$ is an increasing mapping.