Indecomposable ordinal

The additively indecomposable ordinals are exactly the powers ${\omega }^{\delta }$.

A nonzero ordinal $\delta$ is additively closed when $\beta +\gamma <\delta$ for all $\beta ,\gamma <\delta$. These are exactly the ordinals ${\omega }^{\alpha }$: sufficiency follows from Cantor normal form, while necessity follows by splitting any leading coefficient or nonzero tail below $\delta$ into two smaller summands.

For every ordinal $\gamma$ and $\lambda ={\omega }^{\alpha }$, there are unique ordinals $\beta ,\delta$ such thatChoose the greatest initial multiple $\lambda \beta \le \gamma$; if its remainder were at least $\lambda$, the next multiple would still fit. Uniqueness follows because every remainder below $\lambda$ lies before $\lambda (\beta +1)$.

An ordinal $\delta >2$ is multiplicatively closed when $\beta \gamma <\delta$ for all $\beta ,\gamma <\delta$. These are exactly

For nonzero $\lambda$, the ordinal ${\omega }^{\lambda }$ is multiplicatively closed exactly when $\lambda$ is additively closed. One direction follows fromFor the other, the leading exponent of an ordinal product shows that products of ordinals below ${\omega }^{\lambda }$ still have leading exponent below $\lambda$.