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I simply need some more examples for a short teaching session and am out of interesting ideas, any input would be much appreciated (diagonalizable algebras and those which are similar to them are already in my list).

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A class $\mathbb{K}$ of structures (in a fixed language) is elementary - that is, satisfies $\mathbb{K}=\{\mathcal{A}: \mathcal{A}\models T\}$ for some first-order theory $T$ - iff it is closed under isomorphisms, ultraproducts, and ultraroots. Consequently, non-elementary but isomorphism-closed classes form good candidates for classes not closed under ultraproducts. In practice, in fact, I don't know of any naturally-occurring non-elementary class of structures which is ZFC-provably closed under both isomorphisms and under ultraproducts.

Some examples, which you can verify are indeed not closed under ultraproducts:

  • The class of well-orderings.

  • The class of structures of cardinality $<\kappa$ for any fixed infinite cardinal $\kappa$.

  • The class of torsion groups, or - for exactly the same reason - the class of fields of positive characteristic.

  • The class of rigid (= no nontrivial automorphisms) rings. Here "rigid rings" can be replaced by "rigid Xs" for most, but not all, types of X; for example, the only rigid groups are those with $<3$ elements so the class of rigid groups is boringly elementary.

Noah Schweber
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  • I'm curious about your characterisation of elementary classes. Where do the ultraroots come from? I know this if you replace ultraroot by elementary substructure. Do you mean that if $M\preceq N$, then we can find ultrapowers $M^{\mathcal U}\cong N^{\mathcal V}$? (Or maybe even $M^{\mathcal U}\cong N^{\mathcal U}$?) Or is it more complicated? – tomasz Feb 03 '22 at 19:18
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    @tomasz Yup - this is due to Keisler-Shelah (Keisler proved it assuming $\mathsf{GCH}$ and Shelah got a $\mathsf{ZFC}$-only proof; for nuances of the role of cardinal arithmetic here, see e.g. Golshani-Shelah). – Noah Schweber Feb 03 '22 at 19:27
  • Ah, right, I do actually know that one, I guess the "ultraroot" threw me off. – tomasz Feb 03 '22 at 20:47