When I thought whether there is an order-preserving mapping which is also a bijection between two partially ordered sets that have the same cardinality, I got the the answer no.
That's because I found a counter-example, that is there is an infinite descending subsequence in the integer set but not in the natural number set
So my question is
if I have two partially ordered sets that have the same cardinality and both don't have the greatest element or the least element, can I find a order-preserving mapping which is also a bijection between the two sets?
Best Answer
No. Neither $\Bbb Q$ nor $\Bbb Z$ have a greatest or lowest element. But one is dense, while the other is discrete.
Your next question, then, is about density, or so. There it's a bit more complicated, one can state that if both partial orders are linear and dense everywhere, and without minimum or maximum, then they are indeed isomorphic. But otherwise, one can be clever, take two disjoint copies of $\Bbb Q$, or three, or countably many. Take the finite subsets of $\omega$ ordered by inclusion and replace each one with a copy of $\Bbb Q$, or those which have an even number of elements with copies of $\Bbb Q$, and the odd ones with a copy of $\Bbb Z$, or so on.
We can make very complicated partial orders, even if they are all countable. It is generally very hard for two partial orders of the same cardinality to be isomorphic. And you need quite a few limitations.
Some positive results would be:
Well-orders are unique up to isomorphism, but their "discernable traits" run out fairly quickly.
As a consequence, orders which can be partitioned into a well-order and a reversed well-order will also satisfy (at least in thinkable cases) uniqueness up to isomorphism.
The rational numbers are unique up to isomorphism.
The partial order given by the finite subsets of $\Bbb N$ under inclusion is unique, and so is the one given by finite and co-finite subsets.
But by mixing and matching a bunch of them, you can create a great many number of non-isomorphic partial orders. So just going one property at a time is not a good strategy.