Theories coding finite sets

Question

In the paper Weak forms of elimination of imaginaries by Casanovas and Farré, the following notions are considered for a first order theory $T$.

Definition. Let $\bar{M}$ be a big saturated model of $T$, and $M^{eq}$ be its eq-expansion. Then, we say that:

1. $T$ codes finite sets ( $T$ has $\mathrm{FS}$ ) if for every finite set $A$ of real tuples (of the same length) there exists a real tuple $a$ such that for every automorphism $f\in\mathrm{Aut}(\bar{M})$, $f(a) = a$ if and only if $f(A) = A$.
2. $T$ codes Galois finite sets ( $T$ has $\mathrm{GaloisFS}$ ), if for every set of real elements $A$ and for every finite set $C\subseteq\mathrm{acl}(A)$ of tuples (of the same length) there exists a real tuple $a$ such that $\mathrm{Aut}(\bar{M} /Aa ) =\mathrm{Aut}(\bar{M} /A \{C\})$, i.e., for every automorphism $f \in\mathrm{Aut}(\bar{M} /A)$, $f(a) = a$ if and only if $f(C) = C$.

As you can imagine, here $\mathrm{Aut}(\bar{M} /A \{C\})$ denotes the set of all automorphisms of $\bar{M}$ which fix $A$ pointwise and $C$ setwise.

In the same paper, it is pointed out that $\mathrm{FS}\implies \mathrm{GaloisFS}$ and that if $\mathrm{acl}(A)=\mathrm{dcl}(A)$ for every small subset $A$, then $T$ has $\mathrm{GaloisFS}$.

Do you know some examples of theories that have $\mathrm{FS}$, but do not come from Fields Theory context? Or, more interesting, some examples of theories having $\mathrm{GaloisFS}$, but not $\mathrm{FS}$?

Answer 1

Do you know some examples of theories that have FS, but do not come from Fields Theory context?

Any theory which eliminates imaginaries has FS, so you can consider $T^{\mathrm{eq}}$ for your favorite theory $T$.

Another example is DLO, the theory of dense linear orders without endpoints. Here we can give a quite explicit recipe for coding finite sets: Given a finite set of tuples $A = \{\overline{a}_1,\dots,\overline{a}_n\}$, let $\overline{b}$ be the tuple consisting of all elements of the tuples $\overline{a}_i$, listed in increasing order. The point is that if an automorphism fixes $A$ setwise, it must permute the set of coordinates of tuples in $A$, but the only order-preserving permutation of a finite ordered set is the identity.

Or, more interesting, some examples of theories having GaloisFS, but not FS?

The criterion you mentioned in your question, that if $\mathrm{acl} = \mathrm{dcl}$, then $T$ has GaloisFS, gives lots of examples. The theory of an infinite set with no extra structure, the theory of the random graph, the theory of infinite vector spaces over $k$ for some fixed field $k$... these all have $\mathrm{acl} = \mathrm{dcl}$, and hence GaloisFS, but fail to code finite sets.