Can the structure of an ultrafilter determine the structure of its ultrapower?

Question

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

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

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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 $M$, $\prod_{F_1} M\cong \prod_{F_2} M$

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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 $M, N$, $\prod_{F_1} M\cong \prod_{F_2} N$

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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.

Answer 1

With regard to question 1, I don't agree that we usually work with ultrafilters only as pure sets with no structure, for there are all different kinds of ultrafilters, with different properties and features. For example, there is the Rudin-Keisler order on ultrafilers that reveals important fundamental differences, and there is the possibility of p-points, q-points, Ramsey ultrafilters, and so on, even just for ultrafilters on $\omega$. In the large cardinal context of countably complete ultrafilters, we have normal measures, the Mitchell order and many other kinds of structure on the collection of ultrafilters.

Answer 2

I'm not sure what you mean by "notion of structure" either, but there is a lot of work of Andreas Blass from the 70s and 80s dealing with the relationship between ultrafilters on $\omega$ and the associated ultrapowers of the standard model of arithmetic. See for example his survey:

A model-theoretic view of some special ultrafilters. Logic Colloquium '77 (Proc. Conf., Wrocław, 1977), pp. 79–90, Stud. Logic Foundations Math., 96, North-Holland, Amsterdam-New York, 1978.

Answer 3

I am unsure of where you seem to be going at by the notion of structure, but I must mention a certain specific structure sometimes associated with ultrafilters. If $A$ is a set (the set $A$ is usually taken to be infinite), then let $\mathfrak{C}(A)$ be the structure over a language $L_{A}$ with underlying set $A$ such that every function on $A$ is a fundamental operation, every relation is a fundamental relation, and every element of $A$ is a fundamental constant. Let $T_{A}=\mathrm{Th}(\mathfrak{C}(A))$. Then for each ultrafilter $\mathcal{U}$, we associate the ultrapower $\mathfrak{C}(A)^{\mathcal{U}}$. It turns out that each model of $T_{A}$ is a generalized ultrapower of $\mathfrak{C}(A)$ called a limit ultrapower (See Chang and Keisler's book on model theory for a proof of this fact). Furthermore, the finitely generated algebras in $\mathfrak{C}(A)$ are up-to-isomorphism precisely the ultrapowers $\mathfrak{C}(A)^{\mathcal{U}}$ for some ultrafilter $\mathcal{U}$ on a set of cardinality at most $|A|$. The models of $T_{A}$ are very useful when investigating ultrafilters and even different ultrapower constructions since all the information about an ultrafilter $\mathcal{U}$ (or generalized ultrapower construction) is wrapped up in the ultrapower $\mathfrak{C}(A)^{\mathcal{U}}$ for sufficiently large $A$. Suppose that $\mathcal{U},\mathcal{V}$ are ultrafilters on sets of cardinality at most $|A|$. Then $\mathcal{U}$ and $\mathcal{V}$ are Rudin-Keisler equivalent if and only if $\mathfrak{C}(A)^{\mathcal{U}}\simeq\mathfrak{C}(A)^{\mathcal{V}}$ if and only if $\mathcal{A}^{\mathcal{U}}\simeq\mathcal{A}^{\mathcal{V}}$ for all structures $\mathcal{A}$. Furthermore, $\mathcal{U}\leq_{RK}\mathcal{V}$ if and only if $\mathfrak{C}(A)^{\mathcal{U}}$ is elementarily embeddable in $\mathfrak{C}(A)^{\mathcal{V}}$. Similar results hold for direct limits of ultrapowers. Suppose that $(\mathcal{U})_{d\in D},(\mathcal{V})_{e\in E}$ are inverse systems of ultrafilters. Then we have the following for sufficiently large sets $A$: the inverse systems $(\mathcal{U}_{d})_{d\in D},(\mathcal{V}_{e})_{e\in E}$ are isomorphic (in the pro-completion of the category of ultrafilters) if and only if $\varinjlim\mathfrak{C}(A)^{\mathcal{U}_{d}}\simeq\varinjlim\mathfrak{C}(A)^{\mathcal{V}_{e}}$ if and only if $\varinjlim\mathcal{A}^{\mathcal{U}_{d}}\simeq\varinjlim\mathcal{A}^{\mathcal{V}_{e}}$. Of course, in a large cardinal context, there does not seem to be as much of a need to consider ultrapowers of $\mathfrak{C}(A)$ since one can just take ultrapowers the hierarchy of sets $V$.