= Multiplicative closure criterion for a power of omega
For nonzero $\lambda$, the ordinal $\omega^\lambda$ is <multiplicatively closed ordinal>[multiplicatively closed] exactly when $\lambda$ is <additively closed ordinal>[additively closed]. One direction follows from
$$
\omega^\rho\omega^\sigma=\omega^{\rho+\sigma}.
$$
For the other, the <leading exponent of an ordinal product> shows that products of ordinals below $\omega^\lambda$ still have leading exponent below $\lambda$.
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