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

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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

%%comment: dot source: digraph G { node [width = 0.1, height = 0.1] rankdir = BT; rank = same; compound = true; fontname="MathJax_Main"; subgraph cluster_P { node [style=filled,color=white]; edge [arrowhead = "none"]; style = filled; color = lightgrey; fontcolor = black; label = "P"; labelloc = b; a -> b; } subgraph cluster_Q { node [style=filled]; edge [arrowhead = "none"]; color = black; fontcolor = black; label= "Q"; labelloc = b; w -> u; w -> v; u -> t; v -> t; } edge [color = blue, style = dashed] fontcolor = blue; label = "φ"; labelloc = t; b -> u [constraint = false]; a -> v [constraint = false]; } %%

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