Model theoretic Galois correspondence

Question

I am reading the paper “An invitation to model-theoretic Galois theory” and the main theorem is:

If $T$ codes finite sets, that $C = dcl(C)$ is a normal extension of $A = dcl(A)$, and $G := Aut(C/A)$. Then there is a bijection between subgroups of $G$ and intermediate definably closed extensions.

Of course, ACF codes finite sets, and also other theories of fields. Are there other theories where this correspondence is interesting? We know that if the theory has elimination of imaginaries, it codes finite sets. But for $DLO,Th(N,+,*)$(and his variants),$ZFC$ this is corrispondence is trivial (due to rigid automorphism groups and triviality of definible closed sets).

In an another post(I can’t find it anymore) it is pointed out the fact that often the theory fails eliminating imaginaries becomes the theory does not code finite sets. Are there examples of theories that code finite sets but do not eliminate imaginaries?

Related to this question I have another one. Are there some criterion to verify if a theory codes finite sets?

Answer 1

Question 1: I suppose it depends on what you mean by "interesting". There are certainly many examples in which the correspondence is meaningful, in the sense that for a typical definably-closed set $A$, there are many definably closed sets $C$ with $A\subseteq C \subseteq \mathrm{acl}(A)$.

Just staying within the realm of field theory, you could look at the theories of separably closed fields (SCF), pseudo-algebraically closed (PAC) fields, pseudofinite fields, differentially closed fields (DCF), algebraically closed fields with a generic automorphism (ACFA), etc. Outside of field-theoretic examples, you could consider finitely branching trees, or any theory with a definable equivalence relation with infinitely many finite classes, etc. Of course we have to avoid any theory with a definable linear order, since this implies that $\mathrm{acl}(A) = \mathrm{dcl}(A)$ for all sets $A$.

Question 2: A silly answer is to take the theory of an equivalence relation $E$ with infinitely many infinite classes and add all the imaginary sorts necessary to code finite sets. The resulting theory does not have codes for $E$-classes, so it codes finite sets but does not eliminate imaginaries.

Here's a more natural answer: ACVF (the theory of algebraically closed valued fields), or more generally any theory of fields that fails to eliminate imaginaries. Indeed, fields always code finite sets (using the trick with elementary multi-symmetric polynomials. But they need not eliminate imaginaries, e.g. if $K$ is a model of ACVF, the equivalence relation "same coset modulo the maximal ideal" on the valuation ring (whose equivalence classes are the elements of the residue field) is not eliminated.

Question 3: I'm not aware of any general criteria. Coding finite sets is a fairly concrete condition, so to show a theory $T$ codes finite sets, usually you just give the coding explicitly. On the other hand, if you want to show that $T$ does not code finite sets, a common strategy is to find an automorphism $\sigma$ of the monster model that swaps a pair $(a,b)$, i.e., $\sigma(a) = b$ and $\sigma(b) = a$, so that $\sigma$ fixes the unordered pair $\{a,b\}$, and show that $\sigma$ fails to fix any tuple in $\mathrm{dcl}(a,b)$.