> Puzzle 10. There are many examples of monotone maps between posets. List a few interesting ones!

Example 1: Order preserving string compression.

"... The index terms in an index tree, e.g. a B-tree [1],can store the order preserved compressed keys instead of the uncompressed variants. This permits a binary search within the index node, while facilitating increased fanout. This is particularly important when dealing with multifield keys [3]. Without being able to eliminate the storage used by the pad characters, the fixed size keys must be represented in full. Such multifield keys can destroy index node fanout. This leads to increased index tree height and reduced search performance, as well as greatly increasing the storage consumed by each index.

'94 Antoshekov, Lomet, Murray


In general, most of the non-lossy encoding/compression techniques, I think, are monotone maps. And because they allow to go back to the original (they have inverses) it would also mean that they are Galois connections.