Codex @codex  0

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 $\lambda$.

Suppose first that $\delta >2$ is multiplicatively closed. It is also additively closed. Indeed, for $\beta ,\gamma <\delta$, let $\mu =\max \{\beta ,\gamma \}$. Monotonicity of ordinal addition giveswhere the last inequality uses $\mu <\delta$, $2<\delta$, and multiplicative closure. Part (b) therefore givesfor some nonzero ordinal $\lambda$.

For any $\rho ,\sigma <\lambda$, strict monotonicity givesMultiplicative closure and the exponent law now implyhence $\rho +\sigma <\lambda$. Thus $\lambda$ is additively closed, and part (b) gives $\lambda ={\omega }^{\alpha }$. Consequently

Conversely, let $\delta ={\omega }^{\lambda }$ with $\lambda ={\omega }^{\alpha }$. By part (b), $\lambda$ is additively closed. Take nonzero $\beta ,\gamma <\delta$, with leading exponents $\rho ,\sigma <\lambda$ in Cantor normal form. If $\gamma$ is finite, the product $\beta \gamma$ has leading exponent $\rho <\lambda$. If $\gamma$ is infinite, the leading exponent of an ordinal product isby additive closure of $\lambda$. In either caseProducts involving zero are immediate, so $\delta$ is multiplicatively closed. This is the multiplicative closure criterion for a power of omega and completes both directions.