Is there a bijective mapping $f:\mathbb{N}^2→\mathbb{N}$ that preserves lexicographic order? Is there a bijective mapping $f:\mathbb{N}^2→\mathbb{N}$ that preserves lexicographic order?
I know that it is impossyble for the case $f:\mathbb{R}^2→\mathbb{R}$ ( here Bijection from $\mathbb{R}^n$ to $\mathbb{R}$ that preserves lexicographic order?) but seems  I cant use the same arguments.

No. The order type of $N^2$ under lexicographic ordering is $\omega^2$.
