Monotone function

Let $⟨P,{\le }_P⟩$ and $⟨Q,{\le }_Q⟩$ be posets. Then a function $\varphi :P\to Q$ is said to be monotone (alternatively, order-preserving) if for all $s,t\in P$, $s{\le }_{P}t$ implies $\varphi \left(s\right){\le }_Q\varphi \left(t\right)$.

Positive example

Here is an example of a monotone map $\varphi$ from a poset $P$ to another poset $Q$. Since ${\le }_P$ has two comparable pairs of elements, $(c,a)$ and $(b,a)$, there are two constraints that $\varphi$ must satisfy to be considered monotone. Since $c{\le }_{P}a$, we need $\varphi \left(c\right)=u{\le }_{Q}t=\varphi \left(a\right)$. This is, in fact, the case. Also, since $b{\le }_{P}a$, we need $\varphi \left(b\right)=t{\le }_{Q}t=\varphi \left(a\right)$. This is also true.

Negative example

Here is an example of another map $\varphi$ between two other posets $P$ and $Q$. This map is not monotone, because $a{\le }_{P}b$ while $\varphi \left(a\right)=v{\parallel }_{Q}u=\varphi \left(b\right)$.

Additional material

For some examples of montone functions and their applications, see Monotone function: examples. To test your knowledge of monotone functions, head on over to Monotone function: exercises.