Let $\langle\mathbb Q,\lt\rangle$ be the set of rational numbers with the usual $\lt$ ordering.
Find subsets $A$ and $B$ of $\mathbb Q$ which, with the usual ordering, are order isomorphic to the following ordinal arthimetic expressions:
(i) $\omega^2+\omega$
(ii) $2.\omega^3$
Idea: So I think I need to find order preserving bijections between the expressions and subsets of Q. I can see that $Z=\{\frac1n:n \in \mathbb N\}$ is order isomorphic to $\omega$, but I don't see how to deal with isomorphism to powers of $\omega$. I'm aware that a way of thinking of $\omega^2$ is as $\omega$ copies of $\omega$
Solution: (i) So I'm looking for $\omega$ many increasing sequences following on the idea in the comments.
Let $F:[0,\infty) \rightarrow [0,1)$ be a linearly piecewise map such that
$F:[0,1] \rightarrow [0,\frac 12)$
$F:[1,2] \rightarrow [\frac 12,\frac 23)$ and
$F:[n,n+1] \rightarrow [{n \over n+1},{n+1 \over n+2})$
Now let $A^i$={$i- \frac 1n: n \in \mathbb N$} for $i \ge 1$
So each $A^i$ is an increasing sequence of order type $\omega$
Let $Z= \bigcup_{i \in \mathbb N}F"(A^i)$
Then Z has order tupe $ \omega^2$ where $F"(A^i)$ is the range of $F$ on $A^i$.
So $A = Z \cup$ {$k:k \ge 1 \land k \in \mathbb N$} is order isomorphic to $\omega^{2}+\omega$
(ii) Let $C^l$ = {$k+l:k \in B,l \in$ {odds}}, then $\bigcup_{k \in {odds}}C^k$ has order type $\omega^3$
and $\bigcup_{k \in {odds}}C^k \cup \bigcup_{k \in {evens}}C^k$ is order isomorphic to $2.\omega^3$
HINT: If $\omega$ is isomorphic to an increasing sequence, and $\omega+\omega=2\cdot\omega$ is two increasing sequences one after another, and $\omega^2$ is $\omega$ copies of $\omega$, what would that be?