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\$$—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.