Monotone function

Let $\langle P, \leq_P \rangle$ and $\langle Q, \leq_Q \rangle$ be posets. Then a function $\phi : P \rightarrow Q$ is said to be monotone (alternatively, order-preserving) if for all $s, t \in P$, $s \le_P t$ implies $\phi(s) \le_Q \phi(t)$.

Positive example

Here is an example of a monotone map $\phi$ 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 $\phi$ must satisfy to be considered monotone. Since $c \leq_P a$, we need $\phi(c) = u \leq_Q t = \phi(a)$. This is, in fact, the case. Also, since $b \leq_P a$, we need $\phi(b) = t \leq_Q t = \phi(a)$. This is also true.

Negative example

Here is an example of another map $\phi$ between two other posets $P$ and $Q$. This map is not monotone, because $a \leq_P b$ while $\phi(a) = v \parallel_Q u = \phi(b)$.

Additional material

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