In other words, my question is: Is there any order preserving bijection between $\mathbb{N} $ $\times$ $\mathbb{Z}$ and $\mathbb{Z} $ $\times$ $\mathbb{Z}$, with antilexicographic order:
$$ (a,b)<(c,d)⟺(b<d) \lor (b=d \land a<c) $$
(Two sets are similar if there exists a bijection between them which keeps the order in sets.)
I know that $\mathbb{N} $ and $\mathbb{Z} $ are not similar, but still, I don't know what to do with this task...
If there where an order-preserving bijection $f:\mathbb N\times\mathbb Z\to\mathbb Z\times\mathbb Z$ then two consecutive elements would go to two consecutive elements, so $f(a,b)=(c,d)$ implies $f(a+1,b)=(c+1,d)$, this would imply that $\mathbb Z$ and $\mathbb N$ are similar.