Definition:Order Embedding

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

An order monomorphism is a mapping $\phi: S \to T$ such that:


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

Also known as
An order monomorphism is also known as an order embedding.

Also see

 * Order Monomorphism is Injection, which proves that $\phi$ is an injection