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added $Z_2$ example
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user44143
user44143

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

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 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}\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. with automorphism groups ofwhere the automorphisms are other finite permutations, withpermutation groups mixing, or products of those permutations and the more field-theoretic automorphisms, and with $\{x\to x,\ x\to -x\}$groups. Perhaps the cross ratio can be used to get the group of Mobius transformations, if some version of thoseMobius 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.

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

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 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. with automorphism groups of other finite permutations, with groups mixing those permutations and the more field-theoretic automorphisms, and with $\{x\to x,\ x\to -x\}$. Perhaps the cross ratio can be used to get the group of Mobius transformations, if some version of those transformations acts on $\Q$ rather than on $\Q \cup \{\infty\}$.

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.

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

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.

removed Mobius transformations from list
Source Link
user44143
user44143

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

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 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)(c-d)(a'-d')(c'-b') &\\ \phantom{\text{Aut}(} = (a'-b')(c'-d')(a-d)(c-b))& \supseteq \SS{\dfrac{qx+r}{sx+t}:&q,r,s,t\in\Q,\ qt-rs\neq 0}\\ \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}\begin{array}{lll} \A a+b=cd) & =\SS{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. with automorphism groups of other finite permutations, with groups mixing those permutations and the more field-theoretic automorphisms, and with $\{x\to x,\ x\to -x\}$. Perhaps the cross ratio can be used to get the group of Mobius transformations, if some version of those transformations acts on $\Q$ rather than on $\Q \cup \{\infty\}$.

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.

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

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 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)(c-d)(a'-d')(c'-b') &\\ \phantom{\text{Aut}(} = (a'-b')(c'-d')(a-d)(c-b))& \supseteq \SS{\dfrac{qx+r}{sx+t}:&q,r,s,t\in\Q,\ qt-rs\neq 0}\\ \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. with automorphism groups of other finite permutations, with groups mixing those permutations and the more field-theoretic automorphisms, and with $\{x\to x,\ x\to -x\}$.

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.

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

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 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. with automorphism groups of other finite permutations, with groups mixing those permutations and the more field-theoretic automorphisms, and with $\{x\to x,\ x\to -x\}$. Perhaps the cross ratio can be used to get the group of Mobius transformations, if some version of those transformations acts on $\Q$ rather than on $\Q \cup \{\infty\}$.

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.

Source Link
user44143
user44143

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

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 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)(c-d)(a'-d')(c'-b') &\\ \phantom{\text{Aut}(} = (a'-b')(c'-d')(a-d)(c-b))& \supseteq \SS{\dfrac{qx+r}{sx+t}:&q,r,s,t\in\Q,\ qt-rs\neq 0}\\ \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. with automorphism groups of other finite permutations, with groups mixing those permutations and the more field-theoretic automorphisms, and with $\{x\to x,\ x\to -x\}$.

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.