Very specyfic linear order not isomorphic with $\langle\mathbb N,\le\rangle$

Question

Give an example of a numerable linear order such that each element has a successor, there is the smallest element, each element except the smallest one has a predecessor, but the order is not isomorphic with $\langle\mathbb N,\le\rangle$.

I understand it all, but I can't come up with anything.

Answer 1

Consider $\Bbb N$ followed by $\Bbb Z$, i.e., $\{0\}\times \Bbb N\cup \{1\}\times \Bbb Z$ with lexicographic order. Or if you prefer it embedded into the real line: $\{\,\arctan n\mid n\in\Bbb N\,\}\cup\{4+\arctan k\mid k\in\Bbb Z\,\}$.