Is there any order isomorphic function between $\mathbb N \times \mathbb Z$ to $\mathbb Z \times \mathbb N$?

Question

Is there any order isomorphic function between $\mathbb N \times \mathbb Z$ to $\mathbb Z \times \mathbb N$? (With lexicographical order) Thanks

Answer 1

No, there is not. Intuitively, the reason for this is that, while in $\mathbb N\times \mathbb Z$ any element $(a,b)$ is immediately preceded by some $(a,b-1)$ in the order, this is false of $\mathbb Z\times \mathbb N$.

To formalize this, suppose you had an order isomorphism $f:\mathbb N\times \mathbb Z \rightarrow \mathbb Z\times \mathbb N$. Suppose you know $f((a,b))=(0,0)$ for some $(a,b)$, which must hold since $f$ is a bijection. In order to preserve the order $f((a,b-1))$ must be an upper bound to every element strictly less than $f((a,b))$. Conclude from here that no such $f$ exists.