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Let $A,B,C,D$ be algebraic systems and $A$ and $B$ be elementary equivalent as well as $C$ and $D$. Are free products of $A,C$ and $B,D$ elementary equivalent if

  1. $A,B,C,D$ are groups, or
  2. $A,B,C,D$ are Lie algebras?

If one of this problem (or both) has already been solved, it would be nice to get a reference to the corresponding paper(s).

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    $\begingroup$ I assume you already know this, but the answer is positive for direct (Cartesian) products of arbitrary first-order structures. $\endgroup$
    – Emil Jeřábek
    Commented Apr 2, 2019 at 11:31
  • $\begingroup$ If the structures are $\omega$-saturated then you should be able to argue with Ehrenfeucht-Fraïssé games that the resulting free products are elementarily equivalent. This is only non-trivial if the structures are uncountable, though. $\endgroup$
    – James E Hanson
    Commented Apr 2, 2019 at 13:24
  • $\begingroup$ I guess it's meant universal algebraic systems (with only laws, no relations) otherwise I don't see how to define the free products. $\endgroup$
    – YCor
    Commented May 10, 2020 at 21:47
  • $\begingroup$ @EmilJeřábek Do you have a reference? I'm aware of a quite indirect proof, making use of ultraproducts and absoluteness. $\endgroup$
    – YCor
    Commented May 10, 2020 at 21:49
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    $\begingroup$ @YCor Mostowski, On direct products of theories, Journal of Symbolic Logic 17 (1952), 1–31. The Feferman–Vaught theorem is a considerable generalization: Feferman, Vaught, The first order properties of products of algebraic systems, Fundamenta Mathematicae 47 (1959), 57–103. $\endgroup$
    – Emil Jeřábek
    Commented May 11, 2020 at 6:08

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For free products of groups, this is Theorem 7.1 of Zlil Sela's preprint Diophantine geometry over groups X: the elementary theory of free products of groups.

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