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> Does there exist a pair of non-isomorphic structures $\mathfrak{A}$ and $\mathfrak{B}$ as well as sets $I$ and $J$ and ultrafilters $\mathcal{U}$ on $I$ and $\mathcal{F}$ on $J$ such that $\mathfrak{A}^I/\mathcal{U}\cong\mathfrak{B}$ and $\mathfrak{B}^J/\mathcal{F}\cong\mathfrak{A}$?
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This question is inspired by the Keisler-Shelah theorem, which says that you can arrange $\mathfrak{A}^I/\mathcal{U} \cong \mathfrak{B}^J/\mathcal{F}$ (moreover you can arrange that $\mathfrak{A}^I/\mathcal{U}\cong \mathfrak{B}^I/\mathcal{U}$). The answer to this question very likely depends on set theoretic assumptions, so a reasonable weakening would be mere consistency relative to some strong set theoretic assumptions or in some forcing extension. Another reasonable weakening would be to ask about iterated ultrapowers rather than just ultrapowers.


On the other hand assuming a positive answer an obvious follow-up would be the question of the existence of a triple of pairwise non-isomorphic structures, each of which is isomorphic to ultrapowers of the other two. Another obvious follow-up is whether or not it can be arranged that $\mathfrak{A}^I/\mathcal{U}\cong\mathfrak{B}$ and $\mathfrak{B}^I/\mathcal{U\cong{\mathfrak{A}}}$, i.e. whether or not there exists a structure and an ultrafilter such that iterating ultrapowers by that ultrafilter alternates between two isomorphism classes.



Is there a pair of non-isomorphic structures each of which is isomorphic to an ultrapower of the other?
An example assuming large cardinals: suppose $U$ and $W$ are normal ultrafilters on a measurable cardinal $\kappa$ such that $(V\_{\kappa+2})^{M\_U}\neq (V\_{\kappa+2})^{M\_W}$, where $M\_U$ denotes the transitive collapse of the ultrapower of $V$. Let $Z = U\times W$. Let $M$ be the iterated ultrapower of length $\omega$ hitting $Z$ and its images. Then $M = j\_Z(M) = j\_U(j\_W(M))$. But $j\_W(M)$ is not isomorphic to $M$ since $(V\_{\kappa+2})^M = (V\_{\kappa+2})^{M\_U}\neq (V\_{\kappa+2})^{M\_W} = (V\_{\kappa+2})^{j\_W(M)}$. For any structure $N$ in the language of set theory and any ultrafilter $D$, $j\_D(N)$ is isomorphic to the (external) ultrapower of $N$ by $D$. So $M^\kappa / W$ is not isomorphic to $M$ but $(M^{\kappa}/W)^\kappa/U$ is isomorphic to $M$, as desired. If you want to replace $M$ with a set sized structure, consider instead $(V\_\lambda)^M$ where $\lambda$ is a fixed point of $j\_Z$.


