Poset of automorphism groups of variants of a structure

Question

Say that two structures $\mathfrak{A},\mathfrak{B}$ in possibly different finite languages are parametrically equivalent - and write "$\mathfrak{A}\approx\mathfrak{B}$" - iff they have the same underlying set and the same set of definable-with-parameters relations. I'm curious about the "automorphism group spectrum" of a parametric equivalence class. Specifically, for a structure $\mathfrak{A}$ let $$\mathsf{AGS}(\mathfrak{A})=\{\operatorname{Aut}(\mathfrak{B}):\mathfrak{B}\approx\mathfrak{A}\}.$$

This family of groups is a poset with respect to inclusion. Here the "same underlying set" condition is crucial: $\mathsf{AGS}(\mathfrak{A})$ is a family of subgroups of the full permutation group of that underlying set, and I am not identifying isomorphic elements of $\mathsf{AGS}(\mathfrak{A})$ or requiring the $\mathfrak{B}$s considered in forming $\mathsf{AGS}(\mathfrak{A})$ to be in the same signature as $\mathfrak{A}$ itself.

There is a lot of potential flexibility here. First of all, even if $\mathfrak{A}$ is pointwise-definable (so that $\operatorname{Aut}(\mathfrak{A})$ is trivial a fortiori) the set $\mathsf{AGS}(\mathfrak{A})$ may be quite rich: in particular, Matt F. showed that the structure $\mathfrak{Q}$ considered below is parametrically equivalent to a structure whose automorphism group acts $1$-transitively. A bit more surprisingly (to me at least), at MSE Harry West showed that $\mathsf{AGS}(\mathfrak{A})$ need not be upwards-directed.

But that's about all I know. In particular, I don't know much at all about computing $\mathsf{AGS}(\mathfrak{A})$ even for relatively simple interesting structures. I'd like to rectify this. To narrow the field (hehe), I'll focus on the following two candidates:

• The field of rationals $\mathfrak{Q}=(\mathbb{Q};+,\times)$.

• The field of reals $\mathfrak{R}=(\mathbb{R};+,\times)$.

Model-theoretically, $\mathfrak{Q}$ is quite wild (e.g. bi-interpretable with the integers) while $\mathfrak{R}$ is quite tame (e.g. decidable and o-minimal). It's not clear to me which of tameness or wildness makes $\mathsf{AGS}$ easier to analyze, but I suspect that at least one of these has some easily-establishable "coarse" properties (originally I just asked about $\mathfrak{Q}$). That said, all of this may be a red herring: as noted above, even pointwise-definability isn't particularly relevant to $\mathsf{AGS}$, so the model-theoretic dividing lines I'm used to might not be relevant at all.

Anyways:

Problem: describe at least one of $\mathsf{AGS}(\mathfrak{Q})$ or $\mathsf{AGS}(\mathfrak{R})$.

I'm especially interested in the question of whether each is upwards-directed.

Answer 1

$\newcommand{\Q}{\mathbb{Q}}$ $\newcommand{\R}{\mathbb{R}}$ $\newcommand{\A}{\text{Aut}(\Q,\,}$ $\newcommand{\SS}[1]{\{x\to #1 \}}$

Here are some interesting examples, which obviously do not constitute a complete description of the $\text{AGS}$ for either structure.

Let $\A a+b=cd)$ be the automorphisms of $\Q$ considered as a structure with the single relation $\{a,b,c,d:a+b=cd\}$.

With that notation, here are some examples of structures which are parametrically equivalent to $(\Q,\,+,\,\cdot)$ with some lower bounds for their automorphism groups.

\begin{array}{lll} \A a+b=cd) & =\SS{x}&\\ \A a+b=cde) & =\SS{\pm x}&\\ \A ab+cd=ef) &\supseteq \SS{qx:&q\in\Q^*}\\ \A (a-b)(c-d)=(e-f)) &\supseteq \SS{x+r:&r\in\Q}\\ \A (a-b)(c-d)=(e-f)(g-h)) &\supseteq \SS{qx+r:&q\in\Q^*,\, r\in\Q}\\ \A (a+b-cd)(a^2\text{-}1)(b^2\text{-}1)(c^2\text{-}1)(d^2\text{-}1)=0) &=\SS{x,&x\to \sigma(x)}\\ \end{array} where $\sigma(x)$ switches $\pm1$ and fixes everything else.

In all of these cases, it is easy to prove that these structures are parametrically equivalent to the usual rationals by showing that addition and multiplication can be defined with appropriate parameters. And it may be easy to show that the given groups are the automorphism groups (and not just the lower bounds), but I don't see it at the moment.

There should be a lot more examples, e.g. where the automorphisms are other finite permutation groups, or products of those and the more field-theoretic groups. Perhaps some version of Mobius transformations acts on $\Q$ rather than on $\Q \cup \{\infty\}$, and a structure with the cross-ratio has that automorphisms group.

For $\R$, the corresponding list begins with $\text{Aut}(\R,\, a+b=cd)$, which is not just the identity but also includes lots of permutations of transcendental elements. The list continues by adding those permutations to the automorphism groups from $\Q$. On the other hand, $\text{Aut}(R,\, a+b\le cd)$ is just the singleton of the identity.