Hi everyone the I know that the Bernstein- Schroeder Thm says that if we have two sets $A,B$, and there exists an injection $f: A\rightarrow B$ and $g: B\rightarrow A$, then there exists a bijection $h: A\rightarrow B$.
So my question is regard to ordered sets. What happen if $f,g$ are ordered preserving mappings, this of course not sufficient to conclude that $A$ and $B$ are isomorphic, for example in the total case immediately one can think in the closed and open interval on the real line to say something. So there exists a sufficient and necessary condition to guarantees that $A$ and $B$ are isomorphic. Thanks in advance
One sufficient (but not necessary) condition is that $A$ and $B$ are well-ordered sets. I don't know of any necessary conditions.