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.

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.