A well order on $\mathbb Z$ that respects addition?

Question

Does there exists any well-ordering on $\mathbb Z$ that respects addidtion that is if

$a < b$ then $a +c < b+c$ for all $c$ in $\mathbb Z$?

Answer 1

No, such a well-order does not exist. Suppose $<$ is any well-order on $\mathbb{Z}$, let $a$ be the smallest element, and $b = a+1$. Then $a < b$ (since $a$ is the smallest element), and if addition would preserve order, we would have

$$a-1 = a + (-1) < b + (-1) = a,$$

contradicting the minimality of $a$.