Should I write these ordinal operations in Cantor normal form?

Question

I want to increment $\omega^2+\omega+n$ by $\omega$

I.e. the operation I require is $f:\{\omega^2\cdot\alpha+\omega\cdot\beta+n\}\times\{\omega\cdot\beta_2\}\mapsto \{\omega^2\cdot\alpha+\omega\cdot(\beta+\beta_2)+n\}$

What's the best way to describe this? Clearly it's not normal ordinary arithmetic:

$\omega+(\omega^2+\omega+1)=\omega^2+\omega\cdot2+1$

One obvious way is to simply write both in Cantor normal form and say to pairwise add the coefficients of matching exponents.

Then I want to transform $\omega+1$ to $\omega^2+\omega$ by increasing the exponents of every term.

but again I don't think I can just write $(\omega+1)\cdot\omega$ so all I can come up with is to say shift the coefficients left in Cantor normal form... actually I think in this last case maybe to write $\omega\cdot(\omega+1)$ might be okay.

Answer 1

I believe you are looking for the natural (or Hessenberg) sum of ordinals. In the natural sum, you simply add the coefficients in the Cantor normal form.

Answer 2

Ordinal addition and multiplication are not commutative. Incrementing $\omega^2+\omega+1$ by $\omega$ results in $\omega^2+\omega+1+\omega = \omega^2+\omega+(1+\omega) =\omega^2+\omega+\omega = \omega^2+\omega*2$

$(\omega+1)\omega$ is $\omega$ copies of $\omega+1$. That is not the same thing as $\omega^2+\omega$, which is equal to $\omega(\omega+1)$