Definition:Order Embedding/Definition 3

Definition
Let $\left({S, \preceq_1}\right)$ and $\left({T, \preceq_2}\right)$ be ordered sets.

Let $\phi: S \to T$ be a mapping.

Then $\phi$ is an order embedding of $S$ into $T$ both of the following conditions hold:


 * $(1): \quad \phi$ is an injection


 * $(2): \quad \forall x, y \in S: x \prec_1 y \iff \phi \left({x}\right) \prec_2 \phi \left({y}\right)$

Also see

 * Equivalence of Definitions of Order Embedding