Definition:Order-Preserving

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

Let $$\phi: S \to T$$ be an injection.

Then $$\phi$$ is order-preserving iff:

$$\forall x, y \in S: x \le_1 y \Longrightarrow \phi \left({x}\right) \le_2 \phi \left({y}\right)$$