When comparing ordered sets among each other, it is sometimes necessary to decide if their orders are similar or in essence "the same" kind of orders. The following definition states more precisely what is meant by that.
Let $(X,\preceq)$ and $(Y,\preccurlyeq)$ be two posets, chains or strictly ordered sets. An order embedding is an injective function $f:X\to Y$ with the property $a\preceq b$ if and only if $f(a)\preccurlyeq f(b)$ for all $a,b\in X.$
The concept of an order embedding is a generalization of a monotonic function known from analysis. Please also note that $f$ must be injective. Otherwise, $f(a)=f(b)$ could be possible, even if $a\neq b,$ in contradiction to the definition of an order embedding.
Proofs: 3 4