For any ultrafilter $\mathcal{U}$ on $\omega$ and any finite $k$ we can construct tensor power $\mathcal{U}^{\otimes k}$ which is ultrafilter on $\omega^k$. Does there exist some natural extension of this construction for the ordinal $\omega^\omega$?

**Edit**: my suggestion:
$$
\mathcal{U}^{\otimes\omega}=\{B\subset\omega^\omega~|~\{k<\omega~|~B\cap\omega^k\in\mathcal{U}^{\otimes k}\}\in\mathcal{U}\}
$$
But is it good idea?

ordinal$\omega^\omega$, which may exist as some sort of ultralimit of $\mathcal U^{\otimes k}$ with $\cal U$ being the ultrafilter used for the limit. $\endgroup$ – Asaf Karagila♦ Jan 27 '20 at 13:39is, in fact, ordinal exponentiation, not cardinal exponentiation. $\endgroup$ – Emil Jeřábek Jan 28 '20 at 7:337more comments