On the Tukey types of Fubini products

We extend the class of ultrafilters $U$ over countable sets for which $U\cdot U{\equiv }_{T}U$, extending several results from \cite{Dobrinen/Todorcevic11}. In particular, we prove that for each countable ordinal $\alpha \ge 2$, the generic ultrafilter $G_{\alpha }$ forced by $P({\omega }^{\alpha })\mathrm{/}{\text{fin}}^{\otimes \alpha }$ satisfy $G_{\alpha }\cdot G_{\alpha }{\equiv }_T{G}_{\alpha }$. This answers a question posed in \cite[Question 43]{Dobrinen/Todorcevic11}. Additionally, we establish that Milliken-Taylor ultrafilters possess the property that $U\cdot U{\equiv }_{T}U$.