Usually we work with ultrafilters as pure sets without any structure.

**Q1.** Is there any important notion of structure on an ultrafilter?

**Q2.** Is there any non-trivial notion of structure on ultrafilters with the following property:

For all ultrafilters $F_1$ and $F_2$ on the same index set $I$,

$F_1\cong F_2$ **iff** For all language $\mathcal{L}$ and for all $\mathcal{L}$- structure $M$, $\prod_{F_1} M\cong \prod_{F_2} M$

**Q3.** Is there any non-trivial notion of structure on ultrafilters with the following property:

For all ultrafilters $F_1$ and $F_2$ on the same index set $I$,

$F_1\cong F_2$ **iff** For all language $\mathcal{L}$ and for all $\mathcal{L}$- structures $M, N$, $\prod_{F_1} M\cong \prod_{F_2} N$

**Q4.** What are the imapacts of positive or negative answers in the above questions on Boolean valued forcing and large cardinal ultraproducts?

**Remark.** By the "notion of structure" I mean a constant language $\mathcal{L_0}$ and an interpretation function defined for each ultrafilter.