Say that structures $\mathfrak{A},\mathfrak{B}$ with the same underlying set are parametrically equivalent iff every primitive relation/function in one is definable (with parameters) in the other. For example, every group $\mathfrak{G}=(G;*,{}^{-1})$ is parametrically equivalent to its "torsor reduct" $\mathfrak{T}_\mathfrak{G}=(G; (x,y,z)\mapsto x*y^{-1}*z)$. Parametrically equivalent structures can still have very different combinatorial properties. In analogy with the notion of vertex transitive graphs, say that a structure $\mathfrak{A}$ is point-transitive iff the natural action of its automorphism group is $1$-transitive - that is, if for every $a,b\in\mathfrak{A}$ there is some $f\in Aut(\mathfrak{A})$ with $f(a)=b$. If $\mathfrak{G}$ is a nontrivial group then $\mathfrak{G}$ is not point-transitive but $\mathfrak{T_G}$ is point-transitive (for each $g\in G$ the map $a\mapsto g*a$ is in $Aut(\mathfrak{T_G}$)).

I'm broadly interested in understanding the information captured by the family of automorphism groups of parametric equivalents of a given structure (see also here). Motivated by the group/torsor example, one question which seems like it should be easy to answer is: which structures have a parametric equivalent whose automorphism group acts $1$-transitively? But even for simple structures things aren't very clear to me. So I'd like to start with a simple example:

Is the field of rationals $\mathfrak{Q}=(\mathbb{Q};+,\times)$ parametrically equivalent to a point-transitive structure?

Per the torsor example above, the answer is affirmative for $(\mathbb{Q};+)$. It's also affirmative for $(\mathbb{Q};+,<)$ and various similar expansions, by the same construction, so the existence of such a parametric equivalent isn't connected to any obvious model-theoretic tameness property. However, multiplication seems to substantially complicate things.

Say that structures $\mathfrak{A},\mathfrak{B}$ with the same underlying set are parametrically equivalent iff every primitive relation/function in one is definable (with parameters) in the other. For example, every group $\mathfrak{G}=(G;*,{}^{-1})$ is parametrically equivalent to its "torsor reduct" $\mathfrak{T}_\mathfrak{G}=(G; (x,y,z)\mapsto x*y^{-1}*z)$.

Parametrically equivalent structures can still have very different combinatorial properties. In analogy with the notion of vertex transitive graphs, say that a structure $\mathfrak{A}$ is point-transitive iff the natural action of its automorphism group is $1$-transitive - that is, if for every $a,b\in\mathfrak{A}$ there is some $f\in Aut(\mathfrak{A})$ with $f(a)=b$. If $\mathfrak{G}$ is a nontrivial group then $\mathfrak{G}$ is not point-transitive (no automorphism can move the identity) but $\mathfrak{T_G}$ is point-transitive (for each $g\in G$ the map $a\mapsto g*a$ is in $Aut(\mathfrak{T_G}$)).

I'm broadly interested in understanding the information captured by the family of automorphism groups of parametric equivalents of a given structure (see also here). Motivated by the group/torsor example, one question which seems like it should be easy to answer is: which structures have a parametric equivalent whose automorphism group acts $1$-transitively? But even for simple structures things aren't very clear to me. So I'd like to start with a simple example:

Is the field of rationals $\mathfrak{Q}=(\mathbb{Q};+,\times)$ parametrically equivalent to a point-transitive structure?

Per the torsor example above, the answer is affirmative for $(\mathbb{Q};+)$. It's also affirmative for $(\mathbb{Q};+,<)$ and various similar expansions, by the same construction, so the existence of such a parametric equivalent isn't connected to any obvious model-theoretic tameness property. However, multiplication seems to complicate things.

Say that structures $\mathfrak{A},\mathfrak{B}$ with the same underlying set are parametrically equivalent iff every primitive relation/function in one is definable (with parameters) in the other. For example, every group $\mathfrak{G}=(G;*,{}^{-1})$ is parametrically equivalent to its "torsor reduct" $\mathfrak{T}_\mathfrak{G}=(G; (x,y,z)\mapsto x*y^{-1}*z)$. Parametrically equivalent structures can still have very different combinatorial properties - for example, no (nontrivial) group has a $1$-transitive automorphism group action, but every group's torsor reduct does have a $1$-transitive group action (for each $g\in G$ the map $a\mapsto g*a$ is in $Aut(\mathfrak{T_G}$)).

I'm broadly interested in understanding the information captured by the family of automorphism groups of parametric equivalents of a given structure (see also here). Motivated by the group/torsor example, one question which seems like it should be easy to answer is: which structures have a parametric equivalent whose automorphism group acts $1$-transitively? But even for simple structures things aren't very clear to me. So I'd like to start with a simple example:

Is there a structure parametrically equivalent to the field of rationals $\mathfrak{Q}=(\mathbb{Q};+,\times)$ whose automorphism group acts $1$-transitively?

Per the torsor example above, the answer is affirmative for $(\mathbb{Q};+)$. It's also affirmative for $(\mathbb{Q};+,<)$ and various similar expansions, by the same construction, so the existence of such a parametric equivalent isn't connected to any obvious model-theoretic tameness property. However, multiplication seems to substantially complicate things.

Say that structures $\mathfrak{A},\mathfrak{B}$ with the same underlying set are parametrically equivalent iff every primitive relation/function in one is definable (with parameters) in the other. For example, every group $\mathfrak{G}=(G;*,{}^{-1})$ is parametrically equivalent to its "torsor reduct" $\mathfrak{T}_\mathfrak{G}=(G; (x,y,z)\mapsto x*y^{-1}*z)$. Parametrically equivalent structures can still have very different combinatorial properties. In analogy with the notion of vertex transitive graphs, say that a structure $\mathfrak{A}$ is point-transitive iff the natural action of its automorphism group is $1$-transitive - that is, if for every $a,b\in\mathfrak{A}$ there is some $f\in Aut(\mathfrak{A})$ with $f(a)=b$. If $\mathfrak{G}$ is a nontrivial group then $\mathfrak{G}$ is not point-transitive but $\mathfrak{T_G}$ is point-transitive (for each $g\in G$ the map $a\mapsto g*a$ is in $Aut(\mathfrak{T_G}$)).

I'm broadly interested in understanding the information captured by the family of automorphism groups of parametric equivalents of a given structure (see also here). Motivated by the group/torsor example, one question which seems like it should be easy to answer is: which structures have a parametric equivalent whose automorphism group acts $1$-transitively? But even for simple structures things aren't very clear to me. So I'd like to start with a simple example:

Is the field of rationals $\mathfrak{Q}=(\mathbb{Q};+,\times)$ parametrically equivalent to a point-transitive structure?

Per the torsor example above, the answer is affirmative for $(\mathbb{Q};+)$. It's also affirmative for $(\mathbb{Q};+,<)$ and various similar expansions, by the same construction, so the existence of such a parametric equivalent isn't connected to any obvious model-theoretic tameness property. However, multiplication seems to substantially complicate things.