A function on a set $X$ to a set $Y$ is order preserving (monotone, isotone) Consequently, one cannot expect that the inverse of a one-to-one order preserving function will always be order preserving. However, if $X$ and $Y$ are chains and $f$ is one-to-one and isotone, then necessarily $f^-$ is isotone.
In this excerpt, taken from General Topology by John Kelley, the author asserts the inverse of an one-to-one isotone function need not be isotone.
How did the author conclude that?
I would appreciate an illustration that might help me visualise what author meant to say.
Suppose $X$ and $Y$ have the same size, $X$ has no comparable elements and $Y$ does have at least two comparable elements. Then any bijection $X\to Y$ will be vacuously monotone, but the inverse will of course not be.
Or, suppose $\diamondsuit$ is the diamond lattice (four elements, unique max and min, two incomparable elements in between) and $\ell$ is a linearly ordered poset of size four. Then a bijection $\diamondsuit\to\ell$ which sends max to max and min to min will be order-preserving, but the inverse won't be because it won't preserve the ordering of the middle two elements.