Ultraproduct of pairwise non-equivalent structures It is easy to see that the ultraproduct of a family of structures over a principal ultrafilter is elementarily equivalent to the structure whose index generates the ultrafilter, i.e. if $U$ is generated by $k \in I$, then $\prod_{i \in I} M_i / U \equiv M_k$. My question is, does the inverse also hold? That is, if $\prod_{i \in I} M_i / U \equiv M_k$ does it follow that $U$ is a non-principal ultrafilter that focuses on k?
Well, in general the answer is no, because we can simply consider the ultrapower of any structure over a non-principal ultrafilter. Clearly that's gonna be elementarily equivalent to that structure but the ultrafilter can still be non-principal.
Does the same hold if we consider a family of pairwise elementarily inequivalent structures? So, if $M_i, i \in I$ is an infinite family of structures such that $M_i \not\equiv M_j$ for $i\neq j$, does there exist a non-principal ultrafilter $U$ such that $\prod_{i \in I} M_i / U \equiv M_k$ for some $k$, or does $U$ have to necessarily be principal for this to hold?
 A: We can perform the following sort of silly trick. If $M_i$ is a family of pairwise inequivalent structures ranging over an index set $I$ and $U$ is a non-principal ultrafilter on $I$, write $M_{\infty} = \prod_{i \in I} M_i/U$ for the ultraproduct. There are two cases:
Case 1: $M_{\infty}$ is elementary equivalent to some $M_i$. In this case we are done.
Case 2: $M_{\infty}$ is not elementary equivalent to any of the $M_i$. Then we can extend $M_i$ to a family of structures ranging over the index set $I_{\infty} = I \cup \{ \infty \}$ which is still pairwise inequivalent. $U$ induces a unique non-principal ultrafilter $U_{\infty}$ on $I_{\infty}$ given by taking all the subsets in $U$ and adding to them all the subsets in $U$ together with $\infty$. Now consider the ultraproduct
$$M_{\infty + 1} = \prod_{i \in I_{\infty}} M_i/U_{\infty}.$$
By Łoś's theorem, $M_{\infty+1} \models \phi$ iff $X_{\infty} = \{ i \in I_{\infty} : M_i \models \phi \} \in U_{\infty}$. By construction, this is true iff $X = \{ i \in I : M_i \models \phi \} \in U$, which is true iff $M_{\infty} \models \phi$ by a second application of Łoś's theorem. So $M_{\infty+1}$ is elementary equivalent to $M_{\infty}$.
