Properties of natural sum and product of ordinals

Question

There seems to be no publicly available listing of properties of natural (or Hessenberg) addition and multiplication of ordinals. So I'm trying to make one. Please confirm correctness of the following properties or provide corrections, and provide any additional relevant properties.

Recall the definitions: Let $\alpha=\omega^{\alpha_1} + \cdots + \omega^{\alpha_k}$ and $\beta = \omega^{\beta_1}+\cdots+\omega^{\beta_\ell}$ be in CNF (so $\alpha_1\ge \cdots\ge\alpha_k$ and $\beta_1\ge\cdots\ge\beta_\ell$). Let $\gamma_1,\ldots, \gamma_{k+\ell}$ be $\alpha_1,\ldots,\alpha_k,\beta_1,\ldots,\beta_\ell$ sorted in nonincreasing order. Then the natural sum of $\alpha$ and $\beta$ is defined as $\alpha\oplus\beta = \omega^{\gamma_1}+\cdots+\omega^{\gamma_{k+\ell}}$. And the natural product of $\alpha$ and $\beta$ is defined as $\alpha\otimes\beta = \bigoplus_{1\le i\le k, 1\le j\le\ell} \omega^{\alpha_i\oplus\beta_j}$.

Properties

Natural sum and product are commutative and associative. Natural product distributes over natural sum. Meaning:

• $\alpha\oplus \beta=\beta\oplus\alpha$.
• $\alpha\oplus(\beta\oplus\gamma)=(\alpha\oplus\beta)\oplus\gamma$.
• $\alpha\otimes\beta=\beta\otimes\alpha$.
• $\alpha\otimes(\beta\otimes\gamma)=(\alpha\otimes\beta)\otimes\gamma$.
• $\alpha\otimes(\beta\oplus\gamma)=\alpha\otimes\beta\oplus\alpha\otimes\gamma$.

Monotonicity:

• If $\alpha<\beta$ then $\alpha\oplus\gamma<\beta\oplus\gamma$.
• If $\alpha\le\beta$ then $\alpha\otimes\gamma\le\beta\otimes\gamma$.
• If $\alpha<\beta$ and $\gamma>0$ then $\alpha\otimes\gamma<\beta\otimes\gamma$.

(For the next properties, recall that if $\alpha=\omega^{\alpha_1}+\cdots+\omega^{\alpha_k}$ is in CNF, then $\alpha\omega = \lim_n\alpha n = \omega^{\alpha_1+1}$, and $\alpha^\omega = \lim_n \alpha^n = \omega^{\alpha_1\omega}$.)

• $\alpha+\beta\le\alpha\oplus\beta$.
• $\alpha\beta\le\alpha\otimes\beta$.
• $\underbrace{\alpha\oplus\cdots\oplus\alpha}_n=\alpha\otimes n$.
• $\lim_n \alpha\oplus n = \alpha+\omega$. (Not $\alpha\oplus\omega$!)
• If $\alpha$ and $\beta$ are limit ordinals, then $\alpha\oplus\beta = \lim_{\alpha'<\alpha,\beta'<\beta} \alpha'\oplus\beta'$.
• $\lim_n \alpha\otimes n = \alpha\omega$. (Not $\alpha\otimes\omega$!)
• $\lim_n \underbrace{\alpha\otimes\cdots\otimes\alpha}_n = \alpha^\omega$.

Repeated natural product: Recall that every ordinal $\gamma$ can be uniquely decomposed into $\gamma=\omega\beta+n$ for some $\beta$ and some $n$. Then, if we write $\alpha^{[n]}$ to denote $\underbrace{\alpha\otimes\cdots\otimes\alpha}_n$, then it would make sense to write $\alpha^{[\omega\beta+n]}$ to denote $(\alpha^\omega)^\beta\otimes\underbrace{\alpha\otimes\cdots\otimes\alpha}_n$.

Upper bounds:

• If both $\alpha<\omega^\gamma$ and $\beta<\omega^\gamma$ then $\alpha\oplus\beta<\omega^\gamma$.
• If both $\alpha<\omega^{\omega^\gamma}$ and $\beta<\omega^{\omega^\gamma}$ then $\alpha\otimes\beta<\omega^{\omega^\gamma}$.

Answer 1

I covered a bunch of this in my paper Intermediate arithmetic operations on ordinal numbers (arXiv version); see in particular Tables 2 and 3. That's not necessarily everything there is to say on the matter, but I think it covers at least the bulk of what you're asking for!

(Putting this answer up quickly, maybe will come back and edit in more later if needed.)

Answer 2

Now I see that de Jongh and Parikh also define the "repeated natural product". They do so inductively by letting $\alpha^{[0]}=1$, $\alpha^{[\beta+1]}=\alpha^{[\beta]}\otimes\alpha$, and $\alpha^{[\beta]}=\lim_{\gamma<\beta} \alpha^{[\gamma]}$ for limit $\beta$ (Definition 3.6 in their paper). They prove that $(\omega^{\omega^\alpha})^{[\beta]}=(\omega^{\omega^\alpha})^\beta$ for arbitrary $\beta$, whereas what I wrote in the question implies that $\alpha^{[\beta]}=\alpha^\beta$ for limit $\beta$ and arbitrary $\alpha$.

If I find further properties (e.g. in the literature suggested by Dave L Renfro) I will update this answer.