I agree with Patrick #1 on **Puzzle 11** (or have the same confusion!). A simpler example: Let \$$A = \\{a, b\\}\$$ and \$$\leq_A\$$ be only \$$\leq_A = \\{(a, a), (b,b)\\}\$$. Let \$$(B, \leq_B)\$$ be given by \$$B=\\{1, 2\\}\$$ with the usual \$$\leq_B\$$. Then \$$f = \\{(a, 1), (b, 2)\\}\$$ is trivially monotone, but its inverse \$$g\$$ clearly isn't.