A (First-Order) theory T such that every embedding between T-models is elementary

Question

I was wondering, is there a characterization for such a theory? It seems like a pretty handy property so there must be something. Would really appreciate any insight.

Answer 1

This property is called model completeness and it is well-studied. It is easy to see that if every sentence is equivalent mod $T$ to a universal sentence, then every embedding of models of $T$ is elementary (or if equivalently, every sentence is equivalent to an existential sentence). It turns out, via a compactness argument, that this is also necessary, so this is a convenient characterization of model-complete theories. It follows immediately that any theory with quantifier elimination is model-complete. The converse doesn't hold.

It's worth noting that despite the name, completeness and model completeness are incomparable properties. For instance, the theory of $(\mathbb N,+,\cdot)$ is complete by definition, but not model-complete, since it is pretty clear that not every sentence is not equivalent to a universal one. Likewise, the theory of dense linear orders with endpoints is complete but not model complete (the first nontrivial embedding you can think of is probably not elementary). On the other hand, the theory of algebraically closed fields has quantifier elimination, but it is not complete until you specify the characteristic. There are simpler examples as well. Like if we have a language with a single predicate symbol, the theory of one-element models is not complete (since we do not specify whether the predicate is satisfied), but any embedding of one-element models is an isomorphism.