Is $\mathbb{Z \times Q }$ with lexicographical order isomorphic to $\mathbb{Z \times Z }$ with lexicographical order?

Question

I think that $\{\mathbb{Z \times Q}, \leq_{lex}\}$ is not isomorphic to $\{\mathbb{Z \times Z }, \leq_{lex}\}$, but I am not sure if my argumentation is sufficient. Is it enough to state that $\{\mathbb{Z \times Q}, \leq_{lex}\}$ is a dense order, while $\{\mathbb{Z \times Z }, \leq_{lex}\}$ is not?

Answer 1

Yes, to prove two totally ordered sets are not isomorphic it is enough to show that one is dense and the other is not. Depending how much detail is required of your proof, though, you may need to actually prove that $\mathbb{Z}\times\mathbb{Q}$ is dense and that $\mathbb{Z}\times\mathbb{Z}$ is not dense, rather than just stating it.