If f is a biyective function then f inverse no necessarily preserve the order?

Question

How can we prove that if two partially ordered sets and f is a bijective function then the inverse of f does not necessarily preserve the order?

Answer 1

Let $f:\{0,1\} -> \{0,1\} $ be such that $f(0)=1$ and $f(1)=0$. Its inverse does not preserve the order.

Answer 2

I suspect you want an example where $f$ is order-preserving. If you take ${\cal X} = (X, \le)$ to be any partially ordered set that is not totally ordered, then you can extend $\le$ to a total order ${\cal X}' = (X, \le')$ (computer scientists call this topological sorting). The identity function on $X$ is then an order-preserving function from ${\cal X}$ to ${\cal X}'$ whose inverse is not order-preserving. The simplest specific example is to take $X = \{0, 1\}$ and then take $\cal X$ to be $X$ ordered so that $0$ and $1$ are incomparable and ${\cal X}'$ to be $X$ ordered with its usual ordering as a subset of $\Bbb{N}$.