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.