Definition:Order Isomorphism

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

Let $$\phi: S \to T$$ be an bijection such that:


 * $$\phi: S \to T$$ is order-preserving;
 * $$\phi^{-1}: T \to S$$ is order-preserving.

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

That is, $$\phi$$ is an order isomorphism iff:

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

So an order isomorphism can be described as a bijection that preserves ordering "in both directions".

Two posets are (order) isomorphic if there exists such an order isomorphism between them.