Fredrick Eisele wrote:

> Related to **Puzzle 11.**. Is it true that for a monotone map f that has an inverse g, there will always exist a monotone map h [not necessarily g]?

I'm assuming you're looking for a monotone map \\(h\\) in the same direction as \\(g\\), that is, if \\(f:A\to B\\) is an invertible monotone map, is there always a monotone map \\(h:B\to A\\)? The answer is yes, but in general \\(h\\) doesn't need to have much to do with \\(f\\) or \\(g\\).

The reason is that as long as \\(A\\) isn't empty, there's always a monotone map \\(h:B\to A\\)â€”just pick any one element of \\(A\\) and send everything there! If \\(A\\) is empty, there aren't any monotone maps \\(B\to A\\) from nonempty \\(B\\)'s, but fortunately, in that case, the existence of the bijective function \\(f:A\to B\\) tells you that \\(B\\) must be empty too. Then the empty function \\(B\to A\\) is monotone, like you want.

> Related to **Puzzle 11.**. Is it true that for a monotone map f that has an inverse g, there will always exist a monotone map h [not necessarily g]?

I'm assuming you're looking for a monotone map \\(h\\) in the same direction as \\(g\\), that is, if \\(f:A\to B\\) is an invertible monotone map, is there always a monotone map \\(h:B\to A\\)? The answer is yes, but in general \\(h\\) doesn't need to have much to do with \\(f\\) or \\(g\\).

The reason is that as long as \\(A\\) isn't empty, there's always a monotone map \\(h:B\to A\\)â€”just pick any one element of \\(A\\) and send everything there! If \\(A\\) is empty, there aren't any monotone maps \\(B\to A\\) from nonempty \\(B\\)'s, but fortunately, in that case, the existence of the bijective function \\(f:A\to B\\) tells you that \\(B\\) must be empty too. Then the empty function \\(B\to A\\) is monotone, like you want.