Example of a monotone mapping:

Let T be the nodes of a tree, ordered by the following relation: \\(x \le y\\) means \\(x\\) is an ancestor of \\(y\\) in the tree.

Let \\(h(n)\\) be the height of the node in the tree, i.e. the number of edges in the path from the root to the node.

Then \\(h: T \rightarrow \mathbb{N}\\) is a monotone mapping.