Examples of a structure that is not closed under taking ultraproducts?

Question

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).

Answer 1

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.