Does $\mathbb{Q}\times\omega_1$ embed into $\omega_1$?

Question

$\mathbb{Q}\times\omega_1$ has the lexicographic ordering.

By embedding, I mean an order preserving map from $\mathbb{Q}\times\omega_1$ to $\omega_1$

Answer 1

No, it does not: it’s not well-ordered, as you can see by considering just the subset $\Bbb Z^-\times\{0\}$.