Alessandro Vignati (Paris Cité University) – What do we know about reduce products?

Fix a sequence of countable structures $\mathcal{M}_n$, for $n$ in $ℕ$, in a given first-order language $\mathcal{L}$. The reduced product of $(\mathcal{M}_n)$, denoted $\prod_n\mathcal{M}_n/\mathrm{Fin}$ is the $\mathcal{L}$-structure obtained by quotienting the product of the $\mathcal{M}_n$ by the equivalence relation of 'eventual equality'. This is similar to the ultraproduct construction, yet its theory is fairly less understood.

We consider the following question: if two reduced products $\prod_n\mathcal{M}_n/\mathrm{Fin}$ and $\prod_n\mathcal{N}_n/\mathrm{Fin}$ are isomorphic, what can be said about relations between structures we started with? Of course, one can obtain isomorphisms by taking fiberwise isomorphisms, and/or shuffling the indexes, but is that all? We discuss how, in specific interesting cases (such as fields, or certain graphs), answers to this question depend on the set theoretic ambient.