Could lexicographic preferences on N^2 be represented by a utility function u: $N^2$ to $Z$? I got this question on a homework:
Could lexicographic preferences on $N^2$ be represented by a utility function u: $N^2$ to $Z$?
I've heard countless times that lexicographic preferences can't be represented by a utility function. But would anything change if u: $N^2$ to $Z$?
If it is a lexicographic preference on $R^2$, the question sounds weird and I don't know if it can be represented by a utility function from $N^2$ to $R$. The domain of the utility should be the same as the domain of the preference relation, otherwise they cannot represent each other for sure.
Now suppose that, you mean, we have a preference on $R^2$, but we are only taking some data from the $\mathbb R^2$, for example only those data in $N^2$, and see that if we could represent those $N^2$ data by a utility defined on $N^2$. This way, I think the problem becomes more interesting and meaningful.  (Ok, so now the question is edited this way.)
If it a lexicographic preference on $N^2$, I think it can be represented by a utility function from $N^2$ to $R$.
The proof should be obvious in this case. A simple construction suffices.
If you have further questions on this case, please do let me know in comment.
Now let's think about the case of $u:N^2\to Z$. This problem becomes more complicated. To solve this problem, you first need to get used to the proof that a lexicographic preference on $R^2$ cannot be represented by utility function with $R$ as codomain. By applying the same method, you'll see that a preference on $N^2$ cannot be represented with a utility with $N$ as codomain. Then going from $N$ to $Z$ will not take much efforts.