# Compositions of ordinal numbers

Let $$\mathbf{Ord}$$ denote the class of ordinals. Equipped with its natural (commutative) sum and product, this is a semiring.

Say that a composition on $$\mathbf{Ord}$$ is a function $$\circ: \mathbf{Ord} \times \mathbf{Ord}^{\geq \omega} \longrightarrow \mathbf{Ord}$$, where $$\mathbf{Ord}^{\geq \omega}$$ is the class of transfinite ordinals, such that:

$$(i)$$: For $$\varphi \in \mathbf{Ord}$$, the map $$_{\varphi}\circ: \mathbf{Ord}^{\geq \omega} \longrightarrow \mathbf{Ord}$$ defined by $$_{\varphi}\circ(\xi)=\varphi \circ \xi$$ is strictly increasing, and $$_{\omega}\circ=\operatorname{id_{\mathbf{Ord}^{\geq \omega}}}$$.

$$(ii)$$: For $$\alpha \in \mathbf{Ord}$$ and $$\beta,\gamma \in \mathbf{Ord}^{\geq \omega}$$, we have $$\beta \circ \gamma \in \mathbf{Ord}^{\geq \omega}$$ and $$\alpha \circ (\beta \circ \gamma)=(\alpha \circ \beta) \circ \gamma$$.

$$(iii$$): For $$\xi \in \mathbf{Ord}^{\geq \omega}$$, the map $$\circ_{\xi}: \mathbf{Ord}\longrightarrow \mathbf{Ord}$$ defined by $$\circ_{\xi}(\varphi)=\varphi \circ \xi$$ is a strictly increasing semi-ring morphism with $$\circ_{\omega}=\operatorname{id}_{\mathbf{Ord}}$$.

Thus a composition is a way of seeing each ordinal number as an ordinal function with conditions of compatibility with the semi-ring and order structures.

Are there known compositions on the class of ordinal numbers?

• What is $\bf Ord^{\geq\omega}$ for you? Nov 15, 2018 at 23:42
• @AsafKaragila I just edited with the definition. Nov 16, 2018 at 10:33
• Ah. Because I thought you meant all the infinite sequences of ordinals. Nov 16, 2018 at 10:34
• Yes I only realized now this was already a common notation for that. So, just ordinals! Nov 16, 2018 at 10:36
• Well, that makes (ii) much more reasonable now. Nov 16, 2018 at 10:49

So there is at least a somewhat natural way to define a composition of ordinals. I describe it briefly here. The proof that everything works is tedious.

First define $$\mathbf{\Omega}_0=\omega^{\mathbf{Ord}}$$, and for each ordinal $$0<\alpha$$, define the class $$\mathbf{\Omega}_{\alpha}$$ as the class of common fixed points of the functions $$\Omega_{\beta},\beta<\alpha$$ where $$\Omega_{\beta}$$ is the isomorphism $$(\mathbf{Ord},<) \rightarrow (\mathbf{\Omega}_{\beta},<)$$.

For $$0<\alpha \in \mathbf{Ord}$$, let $$F_{\alpha}$$ denote the isomorphism $$(\mathbf{Ord}^{\geq \omega},<) \rightarrow (\mathbf{\Omega}_{\alpha} \setminus \mathbf{\Omega}_{\alpha+1},<)$$. The function $$F_0$$ is defined by $$F_0(\omega\otimes(1\oplus \beta)\oplus n)=2^n \omega^{\omega \otimes(1\oplus f(\beta))}$$ for all $$\beta \in \mathbf{Ord}$$ and $$n<\omega$$, where $$\oplus$$ and $$\otimes$$ are the ordinal sum and product and $$f$$ is the isomorphism $$(\mathbf{Ord},<) \rightarrow (\mathbf{Ord} \setminus \mathbf{\Omega}_1,<)$$. All functions $$F_{\alpha}$$ are strictly increasing with $$\forall \alpha<\beta \in \mathbf{Ord},\forall \xi \in \mathbf{Ord}^{\geq \omega}, \xi.

We also have $$F_0(\xi+\psi)=F_0(\xi) F_0(\psi)$$ and $$F_0(\xi+n)=2^nF_0(\xi)$$ for all ordinals $$\xi,\psi \geq \omega$$ and $$n<\omega$$.

Every ordinal $$\alpha$$ can be uniquely written as $$\alpha=P_0(\omega) \alpha_0 +...+P_r(\omega) \alpha_r$$ where $$r \in \mathbb{N}$$, $$\alpha_0>...>\alpha_r$$ are ordinals of the form $$\alpha_i=\omega^{\beta_i}$$ where $$\beta_i$$ is a limit ordinal, and $$P_0,...,P_r \in \mathbb{N}[X]$$. In this notation, each ordinal $$\alpha_i$$ for $$0 \leq i \leq r$$ with $$\alpha_i>1$$ can be written uniquely as $$F_{\nu_{\alpha_i}}(\alpha_i')$$ for certain $$\nu_{\alpha_i} \in \mathbf{Ord}$$ and $$\alpha_i' \in \mathbf{Ord}^{\geq \omega}$$. We define $$\alpha \circ \beta$$ inductively as $$\alpha \circ \beta = \sum \limits_{\alpha_i>1} P_i(\beta)F_{\nu_{\alpha_i}}(\alpha_i' \circ \beta)+ \sum \limits_{\alpha_r=1} P_r(\beta)$$.

This makes $$\circ$$ the unique composition of ordinals such that for $$\alpha \in \mathbf{Ord}$$ and $$\xi \in \mathbf{Ord}^{\geq \omega}$$, we have $$F_{\alpha}(\omega) \circ \xi = F_{\alpha}(\xi)$$.

For the general method to work, the function $$F_0$$ should transform sums into products and the ranges of the other functions $$F_{\alpha}$$ should be mutually disjoint subclasses of the class of ordinals $$\omega^{\beta}$$ where $$0<\beta$$ is limit. But there still is some freedom in the choice of functions $$F_{\alpha}$$. For instance one can make it so each function $$F_{\alpha+1}$$ is a solution of the Abel equation $$\forall \omega\leq\xi,F_{\alpha+1}(\xi+1)=F_{\alpha}(F_{\alpha+1}(\xi))$$.