Let and be posets. Then a function is said to be monotone (alternatively, order-preserving) if for all , implies .

Positive example

A simple monotone map phi

Here is an example of a monotone map from a poset to another poset . Since has two comparable pairs of elements, and , there are two constraints that must satisfy to be considered monotone. Since , we need . This is, in fact, the case. Also, since , we need . This is also true.

Negative example

A simple, non-monotone map

Here is an example of another map between two other posets and . This map is not monotone, because while .

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.