I am dealing with ordering of rings, so the definition I am using is:
Order Preserving - Let $R'$ and $R$ be ordered rings, and let $P'$, $P$ be their sets of positive elements respectively. Let $f : R'\to R$ be an embedding. We shall say that $f$ is order preserving if for every $x\in R'$ such that $x\in P'$ we have $f(x)\in P$. In other words, $f^{-1}(P)=P'$.
Now the question is: let $K$ be an ordered field and $f : \mathbf{Q}\to K$ be the embedding (why 'the' not just 'an'?) of the rational numbers into K. Show that $f$ is necessarily order preserving.
Anyone have any ideas or solutions on this problem? Thanks a lot.