Definition:Order Embedding

Let $$\left({S; \le_1}\right)$$ and $$\left({T; \le_2}\right)$$ be posets.

Let $$\phi: \left({S; \le_1}\right) \to \left({T; \le_2}\right)$$ be a mapping such that:


 * 1) $$\phi$$ is injective, and
 * 2) $$\forall x, y \in S: x \le_1 y \iff \phi \left({x}\right) \le_2 \phi \left({y}\right)$$, that is, is order-preserving.

Then $$\phi$$ is an order monomorphism.