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.