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Let $M_i_iin I$ be a family of $L$-structures such that for each $iin I$, $T_i=Th(M_i)$ is X where $Xintextstable, simple, NIP, NSOP, dots$. Let $U$ be a non-principal ultrafilter on $I$, and $M=prod_U M_i$. Also, let $T=Th(M)$.

1. Is $T$ an X theory as well?




2. What can we say about $T$ in general?

I think, in general, it would be hard to say somethings about $T$. For example, finite linear orders satisfy stability (and hence simplicity). However a non principal ultraproduct of finite linear orders (except the trivial case) has the strict order property (SOP) and hence is not simple (and in particular, is not stable).

As already noted by others, for most “Shelah-style” dividing lines the answers to the questions are “no” and “probably not much”, respectively.

But I think this question is still nice for fueling interesting examples. For instance, here is an example of the “opposite kind” where the ultra product has better behavior.

Let $M_n$ be the Fraisse limit of finite $K_n$-free graphs, for $ngeq3$. Then $Th(M_n)$ is SOP3 for any $n$. On the other hand, the theory of a nonprincipal ultraproduct of $M_n_ngeq 3$ is that of the random graph. So this theory is simple.

In this example, of course the point is that SOP3 in the individual structures is not witnessed uniformly by the same formula.

EDIT: Here is a proof of the claim above. Let $M=prod_UM_n$ where $U_n$ is nonprincipal. We show that $M$ satisfies the axiomatization of the theory of the random graph. Clearly, $M$ is a graph so we just need to verify the extension axioms. In particular, fix $kgeq 1$ and let $phi$ be the axiom saying that for any two disjoint sets of $k$ vertices, there is some element connected to everything in the first set and nothing in the second set. Then $M_nmodels phi$ for any $n>k+1$, and so $Mmodelsphi$.