Definition:Order Embedding

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

Let $\phi: \left({S, \preceq_1}\right) \to \left({T, \preceq_2}\right)$ be a mapping such that:


 * $\forall x, y \in S: x \ \preceq_1 \ y \iff \phi \left({x}\right) \ \preceq_2 \ \phi \left({y}\right)$

Then $\phi$ is called an order monomorphism.

From Order Monomorphism is Injection it is the case that $\phi$ is an injection.