# Relativized Galois groups of first order theories over a hyperimaginary

Hyoyoon Lee† ∗Department of Mathematics, Yonsei University
50 Yonsei-Ro, Seodaemun-Gu
Seoul, 03722, South Korea<EMAIL_ADDRESS>and  Junguk Lee∗
∗∗Department of Mathematics, Changwon National University
20 Changwondaehak-ro, Uichang-gu
Changwon-si, Gyeongsangnam-do, 51140, South Korea<EMAIL_ADDRESS>
###### Abstract.

We study relativized Lascar groups, which are formed by relativizing Lascar
groups to the solution set of a partial type $\Sigma$. We introduce the notion
of a Lascar tuple for $\Sigma$ and by considering the space of types over a
Lascar tuple for $\Sigma$, the topology for a relativized Lascar group is
(re-)defined and some fundamental facts about the Galois groups of first-order
theories are generalized to the relativized context.

In particular, we prove that any closed subgroup of a relativized Lascar group
corresponds to a stabilizer of a bounded hyperimaginary having at least one
representative in the solution set of the given partial type $\Sigma$. Using
this, we find the correspondence between subgroups of the relativized Lascar
group and the relativized strong types. We also compare the relativized notion
with the restricted one, and provide a condition when two notions coincide.

D. Lascar introduced the notion of Lascar (Galois) group of a first-order
theory in his paper [L82], which is proven to have Galois theoretic
correspondence between the subgroups of the automorphism group and the notions
of model theoretic strong types (cf. [CLPZ01, LP01]). That is, the Lascar, KP,
and Shelah strong types, which play crucial roles in the study of first-order
theories, are exactly the the same as the orbit equivalence relations defined
by the corresponding subgroups of the Lascar group.

In [DKL17], the relativized Lascar group, which are formed by restricting the
domain of the automorphisms that fix the Lascar strong types in a parital
type, was introduced to study the first homology groups in model theory. And
then the relativized Lascar group was studied further in [DKKL21], proving
that under some conditions, the relativized Lascar groups of Shelah strong
types of some finite tuples are all isomorphic. In [KL23], the Galois
theoretic correspondence for the Lascar group and the results concerning the
relativized Lascar group in the previous papers were extended and generalized
to the context of hyperimaginaries. There are also studies connecting the
Lascar group and topological dynamics (such as [KPR18]), but this paper does
not delve into that topic.

There are two natural ways to relativize a subgroup of automorphisms to a
partial type $\Sigma$: One is simply by restricting the domains of the
automoprhisms in the subgroup directly (Definition 3.1, Definition 3.3(2)),
and the other one is collecting restricted automorphisms that fixes the
corresponding orbits in the partial type (Definition 1.19, Definition 3.3(1)).
We will focus on the latter definition, re-define the topology of the
relativized Lascar group, and find the correspondence between the subgroups
and the strong types in the given partial type.

Section 1 is dedicated to presenting preliminaries. In Subsection 1.1, we
recall definitions and facts about hyperimaginaries and Lascar groups. In
particular, the characterizations of Lascar, KP, and Shelah strong types over
a hyperimagianry are given. In the follwing Subsection 1.2, we recall the
definition of the relativized Galois groups of first-order theories over a
hyperimaginary.

In Section 2, we develop most of the main results. We first introduce the
notion of a Lascar tuple over a hyperimaginary $\boldsymbol{e}$ for an
$\boldsymbol{e}$-invariant partial type $\Sigma$ and show its existence
(Definition 2.1 and Lemma 2.2). Under the assumption that there is a tuple $c$
of realizations of $\Sigma$ with $\boldsymbol{e}\in\operatorname{dcl}(c)$, we
define a topology $\mathfrak{t}$ on a relativized Lascar group
$\operatorname{Gal}_{\operatorname{L}}(\Sigma,\boldsymbol{e})$ of $\Sigma$,
which is induced by the surjective map from the space of types over a Lascar
tuple. The topology $\mathfrak{t}$ turns out to coincide with the quotient
topology given from the ordinary Lascar group of the theory, and so
$\operatorname{Gal}_{\operatorname{L}}(\Sigma,\boldsymbol{e})$ becomes a
quasi-compact group with the topology $\mathfrak{t}$.

The key result in Section 2 is the Galois theoretic correspondence between a
closed subgroup of
$\operatorname{Gal}_{\operatorname{L}}(\Sigma,\boldsymbol{e})$ (which can be
characterized by its orbit equivalence relation) and a bounded hyperimaginary,
whose set of representatives intersects with the set of tuples of realizations
of $\Sigma$ (Proposition 2.6 and Lemma 2.9). The Lascar tuple plays an
essential role in the proof of this correspondence, Proposition 2.6. As a
corollary, we find a correspondence between the relativized KP[Shelah] strong
type and the closure of the trivial subgroup[the connected component
containing the identity] of the relativized Lascar group (Theorem 2.15 and
Corollary 2.16).

In Section 3, we recall another method of considering automorphisms only in
the solution set of a partial type by restricting the domain, and compare with
the relativized one. We prove that if the orbit equivalence relation induced
from a given subgroup is weaker than having the same Lascar strong type, then
the relativized subgroup of the automorphism group is the same as the smallest
group containing the restricted one and the relativized Lascar group. We also
present an example that the restricted and relativized Lascar groups are the
same but the Lascar group of the theory is not the same with them.

##  1\. Preliminaries

###  1.1. Hyperimaginaries and model theoretic Galois groups

Throughout, let $T$ be a complete first-order theory and $\mathfrak{C}$ be a
monster model of $T$. Recall that an object is said to be small if its size is
less than the degree of strong homogeneity and saturation of $\mathfrak{C}$.
We will always consider small ordinals and cardinals, and small sets, tuples
and models in $\mathfrak{C}$ otherwise stated.

Any reader who does not want to develop the arguments over a hyperimaginary
(to avoid complexity) may assume that the hyperimaginary $\boldsymbol{e}$
throughout this paper to be an emptyset or interdefinable (Definition 1.4)
with an emptyset, but we need basic definitions and facts about
hyperimaginaries to proceed the arguments even when
$\boldsymbol{e}=\emptyset$.

The proofs for basic properties of hyperimaginaries can be found on [C11] and
[K14]. Most of the basic definitions and facts on the Lascar group can be
found on [KL23], [Lee22] and [K14], which collect and generalize the
fundamental results in [Z02], [CLPZ01], [LP01] and more papers, to the context
of hyperimaginaries.

We denote the automorphism group of $\mathfrak{C}$ by
$\operatorname{Aut}(\mathfrak{C})$ and for $A\subseteq\mathfrak{C}$, we denote
the set of automorphisms of $\mathfrak{C}$ fixing $A$ pointwise by
$\operatorname{Aut}_{A}(\mathfrak{C})$.

######  Definition 1.1.

Let $E$ be an equivalence relation defined on $\mathfrak{C}^{\alpha}$ and
$A\subseteq\mathfrak{C}$. Then $E$ is said to be

  1. (1)

finite if the number of its equivalence classes is finite,

  2. (2)

bounded if the number of its equivalence classes is small,

  3. (3)

$A$-invariant if for any $f\in\operatorname{Aut}_{A}(\mathfrak{C})$, $E(a,b)$
if and only if $E(f(a),f(b))$,

  4. (4)

$A$-definable if there is a formula $\varphi(x,y)$ over $A$ such that
$\models\varphi(a,b)$ if and only if $E(a,b)$ holds, just definable if it is
definable over some parameters, and

  5. (5)

$A$-type-definable if there is a partial type $\Phi(x,y)$ over $A$ such that
$\models\Phi(a,b)$ if and only if $E(a,b)$ holds, just type-definable if it is
type-definable over some parameters.

######  Definition 1.2.

Let $E$ be an $\emptyset$-type-definable equivalence relation on
$\mathfrak{C}^{\alpha}$. An equivalence class of $E$ is called a
hyperimaginary and it is denoted by $a_{E}$ if it has a representative $a$. A
hyperimaginary $a_{E}$ is said to be countable if $|a|$ is countable.

To make hyperimaginaries easily distinguishable from real tuples, we will
write hyperimaginaries in boldface.

######  Definition 1.3.

For a hyperimaginary $\boldsymbol{e}$,

$\operatorname{Aut}_{\boldsymbol{e}}(\mathfrak{C}):=\\{f\in\operatorname{Aut}(\mathfrak{C}):f(\boldsymbol{e})=\boldsymbol{e}\text{
(setwise)}\\}.$

We say an equivalence relation $E$ is $\boldsymbol{e}$-invariant if for any
$f\in\operatorname{Aut}_{\boldsymbol{e}}(\mathfrak{C})$, $E(a,b)$ holds if and
only if $E(f(a),f(b))$ holds.

######  Definition 1.4.

For a hyperimaginary $\boldsymbol{e}$, we say two ‘objects’ (e.g. elements of
$\mathfrak{C}$, tuples of equivalence classes, enumerations of sets) $b$ and
$c$ are interdefinable over $\boldsymbol{e}$ if for any
$f\in\operatorname{Aut}_{\boldsymbol{e}}(\mathfrak{C})$, $f(b)=b$ if and only
if $f(c)=c$. We may omit ‘over $\boldsymbol{e}$’ if
$\boldsymbol{e}=\emptyset$.

######  Fact 1.5 ([K14, Section 4.1] or [C11, Chapter 15]).

  1. (1)

Any tuple (of elements) in $\mathfrak{C}$, any tuple of imaginaries of
$\mathfrak{C}$, and any tuple of hyperimaginaries are interdefinable with a
single hyperimaginary.

  2. (2)

Any hyperimaginary is interdefinable with a sequence of countable
hyperimaginaries.

Until the end of this paper, we fix some arbitrary $\emptyset$-type-definable
equivalence relation $E$ and a hyperimagianry $\boldsymbol{e}:=a_{E}$.

######  Definition 1.6.

  1. (1)

A hyperimaginary $\boldsymbol{e}^{\prime}$ is definable over $\boldsymbol{e}$
if $f(\boldsymbol{e}^{\prime})=\boldsymbol{e}^{\prime}$ for any
$f\in\operatorname{Aut}_{\boldsymbol{e}}(\mathfrak{C})$.

  2. (2)

A hyperimaginary $\boldsymbol{e}^{\prime}$ is algebraic over $\boldsymbol{e}$
if
$\\{f(\boldsymbol{e}^{\prime}):f\in\operatorname{Aut}_{\boldsymbol{e}}(\mathfrak{C})\\}$
is finite.

  3. (3)

A hyperimaginary $\boldsymbol{e}^{\prime}$ is bounded over $\boldsymbol{e}$ if
$\\{f(\boldsymbol{e}^{\prime}):f\in\operatorname{Aut}_{\boldsymbol{e}}(\mathfrak{C})\\}$
is small.

  4. (4)

The definable closure of $\boldsymbol{e}$, denoted by
$\operatorname{dcl}(\boldsymbol{e})$ is the set of all countable
hyperimaginaries $\boldsymbol{e}^{\prime}$ definable over $\boldsymbol{e}$.

  5. (5)

The algebraic closure of $\boldsymbol{e}$, denoted by
$\operatorname{acl}(\boldsymbol{e})$ is the set of all countable
hyperimaginaries $\boldsymbol{e}^{\prime}$ algebraic over $\boldsymbol{e}$.

  6. (6)

The bounded closure of $\boldsymbol{e}$, denoted by
$\operatorname{bdd}(\boldsymbol{e})$ is the set of all countable
hyperimaginaries $\boldsymbol{e}^{\prime}$ bounded over $\boldsymbol{e}$.

######  Definition/Remark 1.7.

  1. (1)

By Fact 1.5(2), if $f\in\operatorname{Aut}(\mathfrak{C})$ fixes
$\operatorname{bdd}(\boldsymbol{e})$ elementwise, then for any hyperimaginary
$\boldsymbol{e}^{\prime}$ which is bounded over $\boldsymbol{e}$,
$f(\boldsymbol{e}^{\prime})=\boldsymbol{e}^{\prime}$. Similar statements also
hold for $\operatorname{dcl}(\boldsymbol{e})$ and
$\operatorname{acl}(\boldsymbol{e})$.

  2. (2)

By (1), for a hyperimaginary $b_{F}$ which is possibly not countable, we write
$b_{F}\in\operatorname{bdd}(\boldsymbol{e})$ if $b_{F}$ is bounded over
$\boldsymbol{e}$. We use notation
$b_{F}\in\operatorname{dcl}(\boldsymbol{e})$,
$\operatorname{acl}(\boldsymbol{e})$ in a similar way.

  3. (3)

Each of $\operatorname{dcl}(\boldsymbol{e})$,
$\operatorname{acl}(\boldsymbol{e})$, and $\operatorname{bdd}(\boldsymbol{e})$
is small and interdefinable with a single hyperimaginary (cf. [C11,
Proposition 15.18]). Thus it makes sense to consider
$\operatorname{Aut}_{\operatorname{bdd}(\boldsymbol{e})}(\mathfrak{C})$.

The complete type of a hyperimaginary over a hyperimaginary, and the equality
of such complete types are type-definable:

######  Definition 1.8 ([K14, Section 4.1]).

Let $b_{F}$ and $c_{F}$ be hyperimaginaries.

  1. (1)

The complete type of $b_{F}$ over $\boldsymbol{e}$,
$\operatorname{tp}_{x}(b_{F}/\boldsymbol{e})$ is a partial type over $a$,

$\exists z_{1}z_{2}(\operatorname{tp}_{z_{1}z_{2}}(ba)\wedge F(x,z_{1})\wedge
E(a,z_{2})),$

whose solution set is the union of automoprhic images of $b_{F}$ over
$\boldsymbol{e}$.

  2. (2)

We write $b_{F}\equiv_{e}c_{F}$ if there is an automorphism
$f\in\operatorname{Aut}_{\boldsymbol{e}}(\mathfrak{C})$ such that
$f(b_{F})=c_{F}$. The equivalence relation $x_{F}\equiv_{e}y_{F}$ in variables
$xy$ is $a$-type-definable, given by the partial type:

$\exists z_{1}z_{2}w_{1}w_{2}(E(a,z_{1})\wedge
E(a,z_{2})\wedge\operatorname{tp}(z_{1}w_{1})=\operatorname{tp}(z_{2}w_{2})\wedge
F(w_{1},x)\wedge F(w_{2},y)).$

We also sometimes use notation
$\operatorname{tp}_{x}(b_{F}/e)=\operatorname{tp}_{y}(c_{F}/e)$ instead of
$b_{F}\equiv_{e}c_{F}$.

Now we start to recall the model theoretic Galois groups.

######  Definition 1.9.

  1. (1)

$\operatorname{Autf_{\operatorname{L}}}(\mathfrak{C},\boldsymbol{e})$ is a
normal subgroup of $\operatorname{Aut}_{\boldsymbol{e}}(\mathfrak{C})$
generated by

$\\{f\in\operatorname{Aut}_{\boldsymbol{e}}(\mathfrak{C}):f\in\operatorname{Aut}_{M}(\mathfrak{C})$
for some $M\models T\text{ such that
}\boldsymbol{e}\in\operatorname{dcl}(M)\\},$

and its elements are called Lascar strong autormophims over $\boldsymbol{e}$.

  2. (2)

The quotient group
$\operatorname{Gal}_{\operatorname{L}}(T,\boldsymbol{e})=\operatorname{Aut}_{\boldsymbol{e}}(\mathfrak{C})/\operatorname{Autf_{\operatorname{L}}}(\mathfrak{C},\boldsymbol{e})$
is called the Lascar group of $T$ over $\boldsymbol{e}$.

######  Fact 1.10 ([KL23, Section 1]).

  1. (1)

$\operatorname{Gal}_{\operatorname{L}}(T,\boldsymbol{e})$ does not depend on
the choice of a monster model up to isomorphism, so it is legitimate to write
$\operatorname{Gal}_{\operatorname{L}}(T,\boldsymbol{e})$ instead of
$\operatorname{Gal}_{\operatorname{L}}(\mathfrak{C},\boldsymbol{e})$.

  2. (2)

$[\operatorname{Aut}_{\boldsymbol{e}}(\mathfrak{C}):\operatorname{Autf_{\operatorname{L}}}(\mathfrak{C},\boldsymbol{e})]=|\operatorname{Gal}_{\operatorname{L}}(T,\boldsymbol{e})|\leq
2^{|T|+|a|}$, which is small.

######  Fact 1.11 ([Z02, Lemma 1]).

Let $M$ be a small model of $T$ such that
$\boldsymbol{e}\in\operatorname{dcl}(M)$ and
$f,g\in\operatorname{Aut}_{\boldsymbol{e}}(\mathfrak{C})$. If
$\operatorname{tp}(f(M)/M)=\operatorname{tp}(g(M)/M)$, then
$f\cdot\operatorname{Autf_{\operatorname{L}}}(\mathfrak{C},\boldsymbol{e})=g\cdot\operatorname{Autf_{\operatorname{L}}}(\mathfrak{C},\boldsymbol{e})$
as elements in $\operatorname{Gal}_{\operatorname{L}}(T,\boldsymbol{e})$.

######  Definition 1.12.

Let $M$ be a small model of $T$ such that
$\boldsymbol{e}\in\operatorname{dcl}(M)$.

  1. (1)

$S_{M}(M)=\\{\operatorname{tp}(f(M)/M):f\in\operatorname{Aut}_{\boldsymbol{e}}(\mathfrak{C})\\}$.

  2. (2)

$\nu:S_{M}(M)\rightarrow\operatorname{Gal}_{\operatorname{L}}(T,\boldsymbol{e})$
is defined by
$\nu(\operatorname{tp}(f(M)/M))=f\cdot\operatorname{Autf_{\operatorname{L}}}(\mathfrak{C},\boldsymbol{e})=[f]$,
which is well-defined by Fact 1.11.

  3. (3)

$\mu:\operatorname{Aut}_{\boldsymbol{e}}(\mathfrak{C})\rightarrow S_{M}(M)$ is
defined by $\mu(f)=\operatorname{tp}(f(M)/M)$.

  4. (4)

$\pi=\nu\circ\mu:\operatorname{Aut}_{\boldsymbol{e}}(\mathfrak{C})\rightarrow\operatorname{Gal}_{\operatorname{L}}(T,\boldsymbol{e})$,
so that $\pi(f)=[f]$ in
$\operatorname{Gal}_{\operatorname{L}}(T,\boldsymbol{e})$.

######  Remark 1.13 ([KL23, Remark 1.9]).

Let $S_{x}(M)=\\{p(x):|x|=|M|$ and $p(x)$ is a complete type over $M\\}$ be
the compact space of complete types. Note that even if
$\boldsymbol{e}\in\operatorname{dcl}(M)$, possibly $a$ is not in $M$, so that
$S_{M}(M)$ is not $\\{p\in
S_{x}(M):\operatorname{tp}(M/\boldsymbol{e})\subseteq p\\}$. But for any small
model $M$ such that $\boldsymbol{e}\in\operatorname{dcl}(M)$, $S_{M}(M)$ is a
closed (so compact) subspace.

###### Proof.

Let

$r(x,a)=\operatorname{tp}(M/\boldsymbol{e})=\exists
z_{1}z_{2}(\operatorname{tp}_{z_{1}z_{2}}(Ma)\wedge x=z_{1}\wedge E(a,z_{2}))$

and $r_{0}(y,M)=\operatorname{tp}(a/M)$. Then

$r^{\prime}(x,M)=\exists y(r(x,y)\wedge r_{0}(y,M))$

has the same solution set as $\operatorname{tp}(M/\boldsymbol{e})$. Thus
$S_{M}(M)=\\{p\in S_{x}(M):r^{\prime}(x,M)\subseteq p\\}$, which is closed. ∎

######  Fact 1.14 ([KL23, Corollary 1.20]).

  1. (1)

We give the quotient topology on
$\operatorname{Gal}_{\operatorname{L}}(T,\boldsymbol{e})$ induced by
$\nu:S_{M}(M)\rightarrow\operatorname{Gal}_{\operatorname{L}}(T,\boldsymbol{e})$.
The quotient topology does not depend on the choice of $M$.

  2. (2)

$\operatorname{Gal}_{\operatorname{L}}(T,\boldsymbol{e})$ is a quasi-compact
topological group.

Now we introduce the notation for two canonical normal subgroups of the Lascar
group and recall the KP and Shelah group.

######  Definition 1.15.

Let $\operatorname{Gal}_{\operatorname{L}}^{c}(T,\boldsymbol{e})$ be the
topological closure of the trivial subgroup of
$\operatorname{Gal}_{\operatorname{L}}(T,\boldsymbol{e})$, and
$\operatorname{Gal}_{\operatorname{L}}^{0}(T,\boldsymbol{e})$ be the connected
component containing the identity in
$\operatorname{Gal}_{\operatorname{L}}(T,\boldsymbol{e})$. Put
$\operatorname{Autf_{\operatorname{KP}}}(\mathfrak{C},\boldsymbol{e})=\pi^{-1}[\operatorname{Gal}_{\operatorname{L}}^{c}(T,\boldsymbol{e})]$
and
$\operatorname{Autf_{\operatorname{S}}}(\mathfrak{C},\boldsymbol{e})=\pi^{-1}[\operatorname{Gal}_{\operatorname{L}}^{0}(T,\boldsymbol{e})]$.

  1. (1)

The KP(-Galois) group of $T$ over $\boldsymbol{e}$ (where KP is an
abbreviation for Kim-Pillay) is

$\operatorname{Gal}_{\operatorname{KP}}(T,\boldsymbol{e}):=\operatorname{Aut}_{\boldsymbol{e}}(\mathfrak{C})/\operatorname{Autf_{\operatorname{KP}}}(\mathfrak{C},\boldsymbol{e}).$

  2. (2)

The Shelah(-Galois) group of $T$ over $\boldsymbol{e}$ is

$\operatorname{Gal}_{\operatorname{S}}(T,\boldsymbol{e}):=\operatorname{Aut}_{\boldsymbol{e}}(\mathfrak{C})/\operatorname{Autf_{\operatorname{S}}}(\mathfrak{C},\boldsymbol{e}).$

Note that we have

$\operatorname{Gal}_{\operatorname{KP}}(T,\boldsymbol{e})\cong\operatorname{Gal}_{\operatorname{L}}(T,\boldsymbol{e})/\operatorname{Gal}_{\operatorname{L}}^{c}(T,\boldsymbol{e}),\
\operatorname{Gal}_{\operatorname{S}}(T,\boldsymbol{e})\cong\operatorname{Gal}_{\operatorname{L}}(T,\boldsymbol{e})/\operatorname{Gal}_{\operatorname{L}}^{0}(T,\boldsymbol{e}).$

The strong types are defined in terms of equivalence relations satisfying
particular conditions, but because we are considering a Lascar group over a
hyperimaginary $\boldsymbol{e}$, it is beneficial to define them as the orbit
equivalence relations and then characterize them.

######  Definition 1.16.

Given hyperimaginaries $b_{F},c_{F}$, they are said to have the same

  1. (1)

Lascar strong type over $\boldsymbol{e}$ if there is
$f\in\operatorname{Autf_{\operatorname{L}}}(\mathfrak{C},\boldsymbol{e})$ such
that $f(b_{F})=c_{F}$, and it is denoted by
$b_{F}\equiv^{\operatorname{L}}_{\boldsymbol{e}}c_{F}$,

  2. (2)

KP strong type over $\boldsymbol{e}$ if there is
$f\in\operatorname{Autf_{\operatorname{KP}}}(\mathfrak{C},\boldsymbol{e})$
such that $f(b_{F})=c_{F}$, and it is denoted by
$b_{F}\equiv^{\operatorname{KP}}_{\boldsymbol{e}}c_{F}$, and

  3. (3)

Shelah strong type over $\boldsymbol{e}$ if there is
$f\in\operatorname{Autf_{\operatorname{S}}}(\mathfrak{C},\boldsymbol{e})$ such
that $f(b_{F})=c_{F}$, and it is denoted by
$b_{F}\equiv^{\operatorname{S}}_{\boldsymbol{e}}c_{F}$.

######  Fact 1.17 ([KL23]).

Let $b_{F}$ and $c_{F}$ be hyperimaginaries.

  1. (1)

$b_{F}\equiv^{\operatorname{L}}_{\boldsymbol{e}}c_{F}$ if and only if for any
$\boldsymbol{e}$-invariant bounded equivalence relation $E^{\prime}$ which is
coarser than $F$, $E^{\prime}(b,c)$ holds;
$x_{F}\equiv^{\operatorname{L}}_{\boldsymbol{e}}y_{F}$ is the finest such an
equivalence relation among them.

  2. (2)

The following are equivalent.

    1. (a)

$b_{F}\equiv^{\operatorname{KP}}_{\boldsymbol{e}}c_{F}$.

    2. (b)

$b_{F}\equiv_{\operatorname{bdd}(\boldsymbol{e})}c_{F}$.

    3. (c)

For any $\boldsymbol{e}$-invariant type-definable bounded equivalence relation
$E^{\prime}$ which is coarser $F$, $E^{\prime}(b,c)$ holds
($x_{F}\equiv^{\operatorname{KP}}_{\boldsymbol{e}}y_{F}$ is the finest such an
equivalence relation among them).

  3. (3)

The following are equivalent.

    1. (a)

$b_{F}\equiv^{\operatorname{S}}_{\boldsymbol{e}}c_{F}$.

    2. (b)

$b_{F}\equiv_{\operatorname{acl}(\boldsymbol{e})}c_{F}$.

    3. (c)

For any $\boldsymbol{e}$-invariant type-definable equivalence relation
$E^{\prime}$ coarser than $F$, if $b_{E^{\prime}}$ has finitely many
conjugates over $\boldsymbol{e}$, then $E^{\prime}(b,c)$ holds.

If $F$ is just =, so that $b_{F}$ and $c_{F}$ are just real tuples, then we
can omit “coarser than $F$”.

The following observation is necessary when we consider an equivalence class
of a type-definable equivalence relation over certain parameters as a
hyperimaginary.

######  Fact 1.18.

Let $F$ be an $\boldsymbol{e}$-invariant type-definable equivalence relation
on $\mathfrak{C}^{\alpha}$. Then there is an $\emptyset$-type-definable
equivalence relation $F^{\prime}$ such that for any
$c\in\mathfrak{C}^{\alpha}$, $c_{F}$ and $(ca)_{F^{\prime}}$ are
interdefinable over $\boldsymbol{e}$ so that we can ‘replace’ an equivalence
class of an $\boldsymbol{e}$-invariant type-definable equivalence relation
with a hyperimaginary.

###### Proof.

Since $F$ is $\boldsymbol{e}(=a_{E})$-invariant, $F$ is type-definable over
$a$, say by $F(x,y;a)$. Then for $p(x)=\operatorname{tp}(a)$, put

$F^{\prime}(xz,yw):=(F(x,y;z)\wedge E(z,w)\wedge p(z)\wedge p(w))\vee xz=yw.$

Then $F^{\prime}$ is the desired one. ∎

###  1.2. Relativized model theoretic Galois groups

From now on, we fix an $\boldsymbol{e}$-invariant partial type $\Sigma(x)$
where $x$ is a possibly infinite tuple of variables. That is, for any
$f\in\operatorname{Aut}_{\boldsymbol{e}}(\mathfrak{C})$, $b\models\Sigma(x)$
if and only if $f(b)\models\Sigma(x)$. We write $\Sigma(\mathfrak{C})$ for the
set of tuples $b$ of elements in $\mathfrak{C}$ such that $b\models\Sigma(x)$.

######  Definition 1.19 (Relativization of automoprhism groups to $\Sigma$).

Let $X\in\\{\operatorname{L},\operatorname{KP},\operatorname{S}\\}$.

  1. (1)

$\operatorname{Aut}_{\boldsymbol{e}}(\Sigma)=\operatorname{Aut}_{\boldsymbol{e}}(\Sigma(\mathfrak{C}))=\\{f\upharpoonright\Sigma(\mathfrak{C}):f\in\operatorname{Aut}_{\boldsymbol{e}}(\mathfrak{C})\\}$.

  2. (2)

For a cardinal $\lambda$,
$\operatorname{Autf}^{\lambda}_{X}(\Sigma,\boldsymbol{e})=$

$\\{\sigma\in\operatorname{Aut}_{\boldsymbol{e}}(\Sigma):\text{ for any of
tuple }b=(b_{i})_{i<\lambda}\text{ where each
}b_{i}\models\Sigma(x_{i})\text{, }b\equiv^{X}_{\boldsymbol{e}}\sigma(b)\\}.$

  3. (3)

$\operatorname{Autf}_{X}(\Sigma,\boldsymbol{e})=$

$\displaystyle\\{\sigma\in\operatorname{Aut}_{\boldsymbol{e}}(\Sigma):$
$\displaystyle\text{ for any cardinal }\lambda\text{ and for any tuple
}b=(b_{i})_{i<\lambda}$ $\displaystyle\text{ with each
}b_{i}\models\Sigma(x_{i})\text{, }b\equiv^{X}_{\boldsymbol{e}}\sigma(b)\\}.$

######  Remark 1.20.

For $X\in\\{\operatorname{L},\operatorname{KP},\operatorname{S}\\}$, it is
easy to check that $\operatorname{Autf}^{\lambda}_{X}(\Sigma,\boldsymbol{e})$
and $\operatorname{Autf}_{X}(\Sigma,\boldsymbol{e})$ are normal subgroups of
$\operatorname{Aut}_{\boldsymbol{e}}(\Sigma)$.

######  Definition 1.21.

  1. (1)

For $X\in\\{\operatorname{L},\operatorname{KP},\operatorname{S}\\}$, for any
cardinal $\lambda$,
$\operatorname{Gal}_{X}^{\lambda}(\Sigma,\boldsymbol{e})=\operatorname{Aut}_{\boldsymbol{e}}(\Sigma)/\operatorname{Autf}_{X}^{\lambda}(\Sigma,\boldsymbol{e})$.

  2. (2)

$\operatorname{Gal}_{\operatorname{L}}(\Sigma,\boldsymbol{e})=\operatorname{Aut}_{\boldsymbol{e}}(\Sigma)/\operatorname{Autf_{\operatorname{L}}}(\Sigma,\boldsymbol{e})$
is the Lascar(-Galois) group over $\boldsymbol{e}$ relativized to $\Sigma$.

  3. (3)

$\operatorname{Gal}_{\operatorname{KP}}(\Sigma,\boldsymbol{e})=\operatorname{Aut}_{\boldsymbol{e}}(\Sigma)/\operatorname{Autf_{\operatorname{KP}}}(\Sigma,\boldsymbol{e})$
is the KP(-Galois) group over $\boldsymbol{e}$ relativized to $\Sigma$.

  4. (4)

$\operatorname{Gal}_{\operatorname{S}}(\Sigma,\boldsymbol{e})=\operatorname{Aut}_{\boldsymbol{e}}(\Sigma)/\operatorname{Autf_{\operatorname{S}}}(\Sigma,\boldsymbol{e})$
is the Shelah(-Galois) group over $\boldsymbol{e}$ relativized to $\Sigma$.

######  Remark 1.22.

  1. (1)

For $X\in\\{\operatorname{L},\operatorname{KP},\operatorname{S}\\}$, if
$[f]=[\operatorname{id}]$ in $\operatorname{Gal}_{X}(T,\boldsymbol{e})$, then
$[f]=[\operatorname{id}]$ in $\operatorname{Gal}_{X}(\Sigma,\boldsymbol{e})$.

  2. (2)

In general,
$\operatorname{Gal}_{\operatorname{L}}^{1}(\Sigma)\neq\operatorname{Gal}_{\operatorname{L}}^{2}(\Sigma)$
(cf. [DKKL21, Example 2.3]).

######  Fact 1.23.

  1. (1)

$\operatorname{Autf_{\operatorname{L}}}(\Sigma,\boldsymbol{e})=\operatorname{Autf}_{\operatorname{L}}^{\omega}(\Sigma,\boldsymbol{e})$.

  2. (2)

$\operatorname{Autf_{\operatorname{KP}}}(\Sigma,\boldsymbol{e})=\operatorname{Autf}_{\operatorname{KP}}^{\omega}(\Sigma,\boldsymbol{e})$.

  3. (3)

$\operatorname{Autf_{\operatorname{S}}}(\Sigma,\boldsymbol{e})=\operatorname{Autf}_{\operatorname{S}}^{\omega}(\Sigma,\boldsymbol{e})$.

###### Proof.

(1): The proof is the same as [DKL17, Remark 3.3] and [Lee22, Proposition 6.3]
over a hyperimaginary, so we omit it.

(2): $\sigma\in\operatorname{Autf_{\operatorname{KP}}}(\Sigma,\boldsymbol{e})$
if and only if for any tuple $b$ of realizations of $\Sigma$,
$b\equiv^{\operatorname{KP}}_{\boldsymbol{e}}\sigma(b)$. But
$\equiv^{\operatorname{KP}}_{\boldsymbol{e}}$ is equivalent to
$\equiv_{\operatorname{bdd}(\boldsymbol{e})}$ (which is type-definable) by
Fact 1.17, thus if
$b^{\prime}\equiv^{\operatorname{KP}}_{\boldsymbol{e}}\sigma(b^{\prime})$ for
any corresponding subtuples $b^{\prime},\sigma(b^{\prime})$ of $b,\sigma(b)$,
which are tuples of finitely many realizations of $\Sigma$, then
$b\equiv^{\operatorname{KP}}_{\boldsymbol{e}}\sigma(b)$ by compactness. Thus
if
$\sigma\in\operatorname{Autf}_{\operatorname{KP}}^{\omega}(\Sigma,\boldsymbol{e})$,
then
$\sigma\in\operatorname{Autf_{\operatorname{KP}}}(\Sigma,\boldsymbol{e})$. The
proof for (3) is the same as (2), replacing
$\operatorname{bdd}(\boldsymbol{e})$ by $\operatorname{acl}(\boldsymbol{e})$.
∎

######  Fact 1.24 ([DKL17, Proposition 3.6] or [Lee22, Proposition 6.5]).

The relativized Lascar group
$\operatorname{Gal}_{\operatorname{L}}(\Sigma,\boldsymbol{e})$ does not depend
on the choice of $\mathfrak{C}$.

##  2\. Topology on relativized model theoretic Galois groups

In this section, we will generalize Fact 1.14 and Fact 1.17 to the relativized
Lascar group. To that end, we will find a topology $\mathfrak{t}$ on the
relativized Lascar group
$\operatorname{Gal}_{\operatorname{L}}(\Sigma,\boldsymbol{e})$, so that

  * •

the group $\operatorname{Gal}_{\operatorname{L}}(\Sigma,\boldsymbol{e})$ is a
quasi-compact group with respect to $\mathfrak{t}$, and

  * •

$\operatorname{Autf_{\operatorname{KP}}}(\Sigma,\boldsymbol{e})=\pi^{\prime-1}[\operatorname{Gal}_{\operatorname{L}}^{c}(\Sigma,\boldsymbol{e})]$
and
$\operatorname{Autf_{\operatorname{S}}}(\Sigma,\boldsymbol{e})=\pi^{\prime-1}[\operatorname{Gal}_{\operatorname{L}}^{0}(\Sigma,\boldsymbol{e})]$,

where
$\pi^{\prime}:\operatorname{Aut}_{\boldsymbol{e}}(\Sigma)\rightarrow\operatorname{Gal}_{\operatorname{L}}(\Sigma,\boldsymbol{e})$
is the natural surjective map, and
$\operatorname{Gal}_{\operatorname{L}}^{c}(\Sigma,\boldsymbol{e})$ and
$\operatorname{Gal}_{\operatorname{L}}^{0}(\Sigma,\boldsymbol{e})$ are the
closure of the trivial subgroup and the connected component containing the
identity of $\operatorname{Gal}_{\operatorname{L}}(\Sigma,\boldsymbol{e})$
with respect to the topology $\mathfrak{t}$, respectively.

For this purpose, we introduce a notion that will replace the role of a small
model when we consider Lascar groups.

######  Definition 2.1.

A small tuple $b$ of realizations of $\Sigma$ is called a Lascar tuple (in
$\Sigma$ over $\boldsymbol{e}$) if
$\operatorname{Autf_{\operatorname{L}}}(\Sigma,\boldsymbol{e})=\\{\sigma\in\operatorname{Aut}_{\boldsymbol{e}}(\Sigma):b\equiv^{\operatorname{L}}_{\boldsymbol{e}}\sigma(b)\\}$.

######  Lemma 2.2.

For any $\boldsymbol{e}$-invariant partial type $\Sigma(x)$, there is a Lascar
tuple.

###### Proof.

By Fact 1.10(2), the set of all Lascar equivalence classes

$C=\\{c_{\equiv^{\operatorname{L}}_{\boldsymbol{e}}}:c\text{ is a countable
tuple of realizations of }\Sigma\\}$

is small, say its cardinality is $\kappa$.

Let $b=(b_{i}:i<\kappa)$ be a small tuple that collects representatives of
Lascar classes in $C$, only one for each class. Then $b$ is the desired one;
by Fact 1.23, it is enough to show that for an automorphism
$f\in\operatorname{Aut}_{\boldsymbol{e}}(\mathfrak{C})$, if
$b\equiv^{\operatorname{L}}_{\boldsymbol{e}}f(b)$, then for any tuple $d$ of
countable realizations of $\Sigma$,
$d\equiv^{\operatorname{L}}_{\boldsymbol{e}}f(d)$. Note that there is
$i<\kappa$ such that $d\equiv^{\operatorname{L}}_{\boldsymbol{e}}b_{i}$. Then
we have

$d\equiv^{\operatorname{L}}_{\boldsymbol{e}}b_{i}\equiv^{\operatorname{L}}_{\boldsymbol{e}}f(b_{i})\equiv^{\operatorname{L}}_{\boldsymbol{e}}f(d)$

where the last equivalence follows from the invariance of
$\equiv^{\operatorname{L}}_{\boldsymbol{e}}$. ∎

######  Remark 2.3.

  1. (1)

Any small tuple of realizations of $\Sigma$ can be extended into a Lascar
tuple.

  2. (2)

Any concatenation of two Lascar tuples is again a Lascar tuple.

Now we fix a Lascar tuple $b$ in $\Sigma$ over $\boldsymbol{e}$, a small model
$M$ with $b\in M$, and $\boldsymbol{e}\in\operatorname{dcl}(M)$. Let

$\displaystyle S_{b}(b)$
$\displaystyle:=\\{\operatorname{tp}(\sigma(b)/b):\sigma\in\operatorname{Aut}_{\boldsymbol{e}}(\Sigma)\\}$
$\displaystyle=\\{\operatorname{tp}(f(b)/b):f\in\operatorname{Aut}_{\boldsymbol{e}}(\mathfrak{C})\\}.$

By the natural restriction map $r:S_{M}(M)\rightarrow S_{b}(b)$, which is
continuous with respect to the logic topology on the type space, we have that
$S_{b}(b)$ is a compact space. Next, Consider a map

$\nu_{b}:S_{b}(b)\rightarrow\operatorname{Gal}_{\operatorname{L}}^{\lambda}(\Sigma,\boldsymbol{e}),\
p=\operatorname{tp}(\sigma(b)/b)\mapsto[\sigma].$

Then it is not hard to check that $\nu_{b}$ is well-defined if
$\boldsymbol{e}\in\operatorname{dcl}(b)$. Indeed, given
$\sigma,\sigma^{\prime}\in\operatorname{Aut}_{\boldsymbol{e}}(\Sigma)$ with
$\sigma(b)\equiv_{b}\sigma^{\prime}(b)$, there is
$\tau\in\operatorname{Aut}_{b}(\Sigma)$ such that
$\tau(\sigma^{\prime}(b))=\sigma(b)$ and so
$\sigma^{-1}\tau\sigma^{\prime}\in\operatorname{Aut}_{b}(\Sigma)$. Since
$\boldsymbol{e}\in\operatorname{dcl}(b)$ and $b$ is a Lascar tuple, we have

$\operatorname{Aut}_{b}(\Sigma)=\operatorname{Aut}_{b\boldsymbol{e}}(\Sigma)\leq\operatorname{Autf_{\operatorname{L}}}(\Sigma,\boldsymbol{e}),$

so we have

$[\operatorname{id}]=[\sigma^{-1}\tau\sigma^{\prime}]=[\sigma]^{-1}[\tau][\sigma^{\prime}]=[\sigma]^{-1}[\sigma^{\prime}]\mbox{,
hence }[\sigma]=[\sigma^{\prime}].$

From now on, we assume that there is a Lascar tuple $b$ in $\Sigma$ over
$\boldsymbol{e}$ such that $\boldsymbol{e}\in\operatorname{dcl}(b)$. Note that
it is equivalent to the condition that there is a small tuple $c$ of
realizations of $\Sigma$ such that $\boldsymbol{e}\in\operatorname{dcl}(c)$;
if there is such a small tuple $c$, by Remark 2.3, we may assume that $c$ is a
Lascar tuple.

######  Remark 2.4.

We have the following commutative diagram of natural surjective maps:

${{\operatorname{Aut}_{\boldsymbol{e}}(\mathfrak{C})}}$${{S_{M}(M)}}$${{\operatorname{Gal}_{\operatorname{L}}(T,\boldsymbol{e})}}$${{\operatorname{Gal}_{\operatorname{KP}}(T,\boldsymbol{e})}}$${{\operatorname{Gal}_{\operatorname{S}}(T,\boldsymbol{e})}}$${{\operatorname{Aut}_{\boldsymbol{e}}(\Sigma)}}$${{S_{b}(b)}}$${{\operatorname{Gal}_{\operatorname{L}}(\Sigma,\boldsymbol{e})}}$${{\operatorname{Gal}_{\operatorname{KP}}(\Sigma,\boldsymbol{e})}}$${{\operatorname{Gal}_{\operatorname{S}}(\Sigma,\boldsymbol{e})}}$$\scriptstyle{\mu}$$\scriptstyle{\xi}$$\scriptstyle{\nu}$$\scriptstyle{r}$$\scriptstyle{\eta_{\operatorname{KP}}}$$\scriptstyle{\xi_{\operatorname{L}}}$$\scriptstyle{\eta_{\operatorname{S}}}$$\scriptstyle{\xi_{\operatorname{KP}}}$$\scriptstyle{\xi_{\operatorname{S}}}$$\scriptstyle{\mu_{b}}$$\scriptstyle{\nu_{b}}$$\scriptstyle{\eta_{\operatorname{KP},\Sigma}}$$\scriptstyle{\eta_{\operatorname{S},\Sigma}}$

Put $\pi_{b}:=\nu_{b}\circ\mu_{b}$ and
$\pi_{\Sigma}:=\pi_{b}\circ\xi=\xi_{\operatorname{L}}\circ\pi$. Note that
$\pi=\nu\circ\mu:\operatorname{Aut}_{\boldsymbol{e}}(\mathfrak{C})\rightarrow\operatorname{Gal}_{\operatorname{L}}(T,\boldsymbol{e})$
and
$\pi_{b}=\pi^{\prime}:\operatorname{Aut}_{\boldsymbol{e}}(\Sigma)\rightarrow\operatorname{Gal}_{\operatorname{L}}(\Sigma,\boldsymbol{e})$.

######  Remark 2.5 (Relativized Galois groups are topological groups).

Consider a topology $\mathfrak{t}_{b}$ on
$\operatorname{Gal}_{\operatorname{L}}(\Sigma,\boldsymbol{e})$ given by the
quotient topology via $\nu_{b}$. Then
$(\operatorname{Gal}_{\operatorname{L}}(\Sigma,\boldsymbol{e}),\mathfrak{t}_{b})$
is a quasi-compact topological group whose topology $\mathfrak{t}_{b}$ is
independent of the choice of a Lascar tuple $b$, so there is no harm to denote
this $\mathfrak{t}_{b}$ by $\mathfrak{t}$.

###### Proof.

The restriction map $r:S_{M}(M)\rightarrow S_{b}(b)$ is a continuous
surjective map between compact Hausdorff spaces, hence a quotient map. Thus
$\nu_{b}:S_{b}(b)\rightarrow\operatorname{Gal}_{\operatorname{L}}(\Sigma,\boldsymbol{e})$
and $\nu_{b}\circ
r:S_{M}(M)\rightarrow\operatorname{Gal}_{\operatorname{L}}(\Sigma,\boldsymbol{e})$
induce the same quotient topology on
$\operatorname{Gal}_{\operatorname{L}}(\Sigma,\boldsymbol{e})$.

But the quotient topology on
$\operatorname{Gal}_{\operatorname{L}}(\Sigma,\boldsymbol{e})$ given by the
natural projection map
$\xi_{\operatorname{L}}:\operatorname{Gal}_{\operatorname{L}}(T,\boldsymbol{e})\rightarrow\operatorname{Gal}_{\operatorname{L}}(\Sigma,\boldsymbol{e})$
is also the same as the above topology, thus the topology of
$\operatorname{Gal}_{\operatorname{L}}(\Sigma,\boldsymbol{e})$ is independent
of the choice $b$ (Fact 1.14) and it is a topological group since a quotient
group of a topological group with quotient topology is a topological group (it
is the same reasoning as [DKL17, Remark 3.4]). ∎

The following proposition is a relativized analogue of [KL23, Proposition
2.3], which have extended [LP01, Lemma 1.9] over a hyperimaginary. The purpose
of Proposition 2.6 is to interpret closed subgroups of
$\operatorname{Gal}_{\operatorname{L}}(\Sigma,\boldsymbol{e})$ using bounded
hyperimaginaries, and it will play a critical role in many arguments until the
end of this section.

######  Proposition 2.6.

Let $H\leq\operatorname{Aut}_{\boldsymbol{e}}(\mathfrak{C})$. The following
are equivalent.

  1. (1)

$\pi_{\Sigma}[H]$ is closed in
$\operatorname{Gal}_{\operatorname{L}}(\Sigma,\boldsymbol{e})$ and
$H=\pi_{\Sigma}^{-1}[\pi_{\Sigma}[H]]$.

  2. (2)

$H=\operatorname{Aut}_{\boldsymbol{e}^{\prime}\boldsymbol{e}}(\mathfrak{C})$
for some hyperimaginary
$\boldsymbol{e}^{\prime}\in\operatorname{bdd}(\boldsymbol{e})$, and one of the
representatives of $\boldsymbol{e}^{\prime}$ is a tuple of realizations of
$\Sigma$.

###### Proof.

The method of proof is the same as the proof of [KL23, Proposition 2.3], but
we use a Lascar tuple $b$ instead of a model $M$.

$(\Rightarrow)$: We have

$\operatorname{Aut}_{b}(\mathfrak{C})=\operatorname{Aut}_{b\boldsymbol{e}}(\mathfrak{C})\leq\xi^{-1}[\operatorname{Autf_{\operatorname{L}}}(\Sigma,\boldsymbol{e})]\leq
H$

and since $\pi_{\Sigma}[H]$ is closed, $\nu_{b}^{-1}[\pi_{\Sigma}[H]]$ is
closed and thus $\\{h(b):h\in H\\}$ is type-definable over $b$. Hence by
[KL23, Proposition 2.2],
$H=\operatorname{Aut}_{b_{F}\boldsymbol{e}}(\mathfrak{C})$ for some
$\emptyset$-type-definable equivalence relation $F$.

We have $b_{F}\in\operatorname{bdd}(\boldsymbol{e})$:
$[\operatorname{Aut}_{\boldsymbol{e}}(\mathfrak{C}):H]=\kappa$ is small since
$\operatorname{Autf_{\operatorname{L}}}(\mathfrak{C},\boldsymbol{e})\leq H$,
and so there is
$\\{f_{i}\in\operatorname{Aut}_{\boldsymbol{e}}(\mathfrak{C}):i<\kappa\\}$
such that
$\operatorname{Aut}_{\boldsymbol{e}}(\mathfrak{C})=\bigsqcup_{i<\kappa}f_{i}\cdot
H$. Then for all $g,h\in\operatorname{Aut}_{\boldsymbol{e}}(\mathfrak{C})$, if
$g\cdot H=h\cdot H$, then $h^{-1}g\in H$ and hence $g(b_{F})=h(b_{F})$.

$(\Leftarrow)$: Say $\boldsymbol{e}^{\prime}=c_{F}$ where $c$ is a tuple of
realizations of $\Sigma$ and $F$ is an $\emptyset$-type-definable equivalence
relation. By Remark 2.3 and Remark 2.5, we may assume that the Lascar tuple
$b$ contains $c$. For $q(x)=\operatorname{tp}(c/\boldsymbol{e})$, because
$\boldsymbol{e}^{\prime}\in\operatorname{bdd}(\boldsymbol{e})$,

$F^{\prime}(z_{1},z_{2}):=(q(z_{1})\wedge q(z_{2})\wedge
F(z_{1},z_{2}))\vee(\neg q(z_{1})\wedge\neg q(z_{2}))$

is an $\boldsymbol{e}$-invariant bounded equivalence relation on
$\mathfrak{C}^{|c|}$. Since $c$ is a tuple in $\Sigma(\mathfrak{C})$, for any
$\sigma\in\operatorname{Autf_{\operatorname{L}}}(\Sigma,\boldsymbol{e})$,
$\sigma(c)\equiv^{\operatorname{L}}_{\boldsymbol{e}}c$ and thus
$F^{\prime}(\sigma(c),c)$ by Fact 1.17(1). Note that $c_{F}=c_{F^{\prime}}$,
so we have
$\xi^{-1}[\operatorname{Autf_{\operatorname{L}}}(\Sigma,\boldsymbol{e})]\leq
H=\operatorname{Aut}_{c_{F}\boldsymbol{e}}(\mathfrak{C})$, hence
$\pi_{\Sigma}^{-1}[\pi_{\Sigma}[H]]=H$. Notice that
$H=\\{f\in\operatorname{Aut}_{\boldsymbol{e}}(\mathfrak{C}):f(c)\models
F(z,c)\\}$. Then $\nu_{b}^{-1}[\pi_{\Sigma}[H]]=\\{p(z^{\prime})\in
S_{b}(b):F(z,c)\subseteq p(z^{\prime})\\}$ where $z\subseteq z^{\prime}$ and
$|z^{\prime}|=|b|$. Thus $\pi_{\Sigma}[H]$ is closed. ∎

######  Definition 2.7.

  1. (1)

For $H\leq\operatorname{Aut}_{\boldsymbol{e}}(\mathfrak{C})$, the relation
$\equiv^{H}$ is an orbit equivalence relation such that for tuples $b,c$ in
$\mathfrak{C}$, $b\equiv^{H}c$ if and only if there is $h\in H$ such that
$h(b)=c$.

  2. (2)

For $H\leq\operatorname{Aut}_{\boldsymbol{e}}(\Sigma)$, we will use the same
notation $\equiv^{H}$ but it is confined to the tuples of realizations of
$\Sigma$.

######  Remark 2.8.

Let $H\leq\operatorname{Aut}_{\boldsymbol{e}}(\Sigma)$ and $c,d$ be tuples of
realizations of $\Sigma$.

  1. (1)

If $H=\operatorname{Autf_{\operatorname{L}}}(\Sigma,\boldsymbol{e})$, then
$c\equiv^{H}d$ if and only if $c\equiv^{\operatorname{L}}_{\boldsymbol{e}}d$.

  2. (2)

If $H=\operatorname{Autf_{\operatorname{KP}}}(\Sigma,\boldsymbol{e})$, then
$c\equiv^{H}d$ if and only if $c\equiv^{\operatorname{KP}}_{\boldsymbol{e}}d$.

  3. (3)

If $H=\operatorname{Autf_{\operatorname{S}}}(\Sigma)$, then $c\equiv^{H}d$ if
and only if $c\equiv^{\operatorname{S}}_{\boldsymbol{e}}d$.

Now we prove that closed subgroups of the relativized Lascar groups are
completely determined by the orbit equivalence relations they define.

######  Lemma 2.9.

Let $H$ be a subgroup of $\operatorname{Aut}_{\boldsymbol{e}}(\Sigma)$
containing $\operatorname{Autf_{\operatorname{L}}}(\Sigma,\boldsymbol{e})$
such that $\pi_{b}[H]$ is a closed subgroup of
$\operatorname{Gal}_{\operatorname{L}}(\Sigma,\boldsymbol{e})$. Then

  1. (1)

For any tuples $c$ and $d$ of realizations of $\Sigma$, $c\equiv^{H}d$ if and
only if for all corresponding subtuples $c^{\prime}$ and $d^{\prime}$ of $c$
and $d$, which are finite tuples of realizations of $\Sigma$,
$c^{\prime}\equiv^{H}d^{\prime}$.

  2. (2)

Given $\sigma\in\operatorname{Aut}_{\boldsymbol{e}}(\Sigma)$, the following
are equivalent.

    1. (a)

$\sigma\in H$.

    2. (b)

$\sigma$ fixes all the $\equiv^{H}$-classes of any tuples of realizations of
$\Sigma$.

    3. (c)

$\sigma$ fixes all the $\equiv^{H}$-classes of any finite tuples of
realizations of $\Sigma$.

###### Proof.

(1): Suppose that for all corresponding subtuples $c^{\prime}$ and
$d^{\prime}$ of $c$ and $d$, which are finite tuple of realizations of
$\Sigma$, $c^{\prime}\equiv^{H}d^{\prime}$. Note that, by Proposition 2.6,
$H^{\prime}:=\pi_{\Sigma}^{-1}[\pi_{b}[H]]=\operatorname{Aut}_{\boldsymbol{e}^{\prime}\boldsymbol{e}}(\mathfrak{C})$
for some hyperimaginary
$\boldsymbol{e}^{\prime}\in\operatorname{bdd}(\boldsymbol{e})$ such that one
of its representatives is a tuple of realizations of $\Sigma$. Then by
commutativity of the diagram in Remark 2.4,
$H=\xi[H^{\prime}]=\operatorname{Aut}_{\boldsymbol{e}^{\prime}\boldsymbol{e}}(\Sigma)$
and so $c\equiv^{H}d$ if and only if
$\operatorname{tp}(c/\boldsymbol{e}^{\prime}\boldsymbol{e})=\operatorname{tp}(d/\boldsymbol{e}^{\prime}\boldsymbol{e})$.
Since
$\operatorname{tp}(c^{\prime}/\boldsymbol{e}^{\prime}\boldsymbol{e})=\operatorname{tp}(d^{\prime}/\boldsymbol{e}^{\prime}\boldsymbol{e})$
for all corresponding subtuples $c^{\prime}$ and $d^{\prime}$ of $c$ and $d$,
which are finite tuples of realizations of $\Sigma$, by compactness, we have
that
$\operatorname{tp}(c/\boldsymbol{e}^{\prime}\boldsymbol{e})=\operatorname{tp}(d/\boldsymbol{e}^{\prime}\boldsymbol{e})$
and so $c\equiv^{H}d$.

(2): It is enough to show that $(c)$ implies $(a)$. Take arbitrary
$\sigma\in\operatorname{Aut}_{\boldsymbol{e}}(\Sigma)$ and suppose $\sigma$
fixes all the $\equiv^{H}$-classes of any finite tuples of realizations of
$\Sigma$. For any subtuple $b^{\prime}$ of $b$, which is a finite tuple of
realizations of $\Sigma$, $b^{\prime}\equiv^{H}\sigma(b^{\prime})$, and so by
(1), $b\equiv^{H}\sigma(b)$. Thus there is $\tau\in H$ such that
$\tau(b)=\sigma(b)$ and so $\tau^{-1}\sigma(b)=b$. Then we have

$\tau^{-1}\sigma\in\operatorname{Aut}_{b}(\Sigma)\leq\operatorname{Autf_{\operatorname{L}}}(\Sigma,\boldsymbol{e})\leq
H,$

and conclude that $\sigma\in\tau H=H$. ∎

######  Definition 2.10.

  1. (1)

$\operatorname{Gal}_{\operatorname{L}}^{c}(\Sigma,\boldsymbol{e})$ is the
topological closure of the trivial subgroup of
$\operatorname{Gal}_{\operatorname{L}}(\Sigma,\boldsymbol{e})$.

  2. (2)

$\operatorname{Gal}_{\operatorname{L}}^{0}(\Sigma,\boldsymbol{e})$ is the
connected component containing the identity in
$\operatorname{Gal}_{\operatorname{L}}(\Sigma,\boldsymbol{e})$.

Note that $\operatorname{Gal}_{\operatorname{L}}^{c}(\Sigma,\boldsymbol{e})$
and $\operatorname{Gal}_{\operatorname{L}}^{0}(\Sigma,\boldsymbol{e})$ are
closed normal subgroups because
$\operatorname{Gal}_{\operatorname{L}}(\Sigma,\boldsymbol{e})$ is a
topological group.

######  Lemma 2.11.

$\operatorname{Autf_{\operatorname{KP}}}(\Sigma,\boldsymbol{e})/\operatorname{Autf_{\operatorname{L}}}(\Sigma,\boldsymbol{e})$
and
$\operatorname{Autf_{\operatorname{S}}}(\Sigma,\boldsymbol{e})/\operatorname{Autf_{\operatorname{L}}}(\Sigma,\boldsymbol{e})$
are closed in $\operatorname{Gal}_{\operatorname{L}}(\Sigma,\boldsymbol{e})$.

###### Proof.

We have

$\nu_{b}^{-1}\eta_{\operatorname{KP},\Sigma}^{-1}[\operatorname{Autf_{\operatorname{KP}}}(\Sigma,\boldsymbol{e})/\operatorname{Autf_{\operatorname{L}}}(\Sigma,\boldsymbol{e})]=\\{\operatorname{tp}(f(b)/b):b\equiv^{\operatorname{KP}}_{\boldsymbol{e}}f(b)\\}.$

By Fact 1.17(2) and the proof of Remark 1.13,
$\equiv^{\operatorname{KP}}_{\boldsymbol{e}}$ is type-definable and there is
an $\boldsymbol{e}$-invariant partial type $\Gamma(x,y)$ over $b$ such that
$\\{\operatorname{tp}(f(b)/b):b\equiv^{\operatorname{KP}}_{\boldsymbol{e}}f(b)\\}=\\{p(x)\in
S_{b}(b):\models\Gamma(x,b)\\}$, implying that
$\operatorname{Autf_{\operatorname{KP}}}(\Sigma,\boldsymbol{e})/\operatorname{Autf_{\operatorname{L}}}(\Sigma,\boldsymbol{e})$
is closed. The proof for
$\operatorname{Autf_{\operatorname{S}}}(\Sigma,\boldsymbol{e})/\operatorname{Autf_{\operatorname{L}}}(\Sigma,\boldsymbol{e})$
is exactly the same; replace KP by S. ∎

######  Notation 2.12.

Let $A$ be a set of hyperimaginaries.

$A\cap\Sigma:=\\{\boldsymbol{e}^{\prime}\in A:$ one of the representatives of
$\boldsymbol{e}^{\prime}$ is a tuple of realizations of $\Sigma\\}.$

######  Lemma 2.13.

$\pi_{b}^{-1}[\operatorname{Gal}^{c}_{\operatorname{L}}(\Sigma,\boldsymbol{e})]=\operatorname{Aut}_{\operatorname{bdd}(\boldsymbol{e})\cap\Sigma}(\Sigma)\text{,
and}$
$\pi_{b}^{-1}[\operatorname{Gal}^{0}_{\operatorname{L}}(\Sigma,\boldsymbol{e})]=\operatorname{Aut}_{\operatorname{acl}(\boldsymbol{e})\cap\Sigma}(\Sigma).$

###### Proof.

By the essentially same proof of [KL23, Proposition 3.12(1)], the assertions
follow using Proposition 2.6 and the fact that
$\operatorname{Gal}^{c}_{\operatorname{L}}(\Sigma,\boldsymbol{e})$ is the
intersection of all closed subgroups containing the identity and
$\operatorname{Gal}^{0}_{\operatorname{L}}(\Sigma,\boldsymbol{e})$ is the
intersection of all closed (normal) subgroups of finite indices in
$\operatorname{Gal}_{\operatorname{L}}(\Sigma,\boldsymbol{e})$. ∎

######  Proposition 2.14.

  1. (1)

Let
$H=\pi_{b}^{-1}[\operatorname{Gal}_{\operatorname{L}}^{c}(\Sigma,\boldsymbol{e})]\leq\operatorname{Aut}_{\boldsymbol{e}}(\Sigma)$.
Then $c\equiv^{H}d$ if and only if
$c\equiv^{\operatorname{KP}}_{\boldsymbol{e}}d$.

  2. (2)

Let
$H=\pi_{b}^{-1}[\operatorname{Gal}_{\operatorname{L}}^{0}(\Sigma,\boldsymbol{e})]\leq\operatorname{Aut}_{\boldsymbol{e}}(\Sigma)$.
Then $c\equiv^{H}d$ if and only if
$c\equiv^{\operatorname{S}}_{\boldsymbol{e}}d$.

###### Proof.

(1): By Proposition 2.6, we know that
$\pi_{\Sigma}^{-1}[\operatorname{Gal}_{\operatorname{L}}^{c}(\Sigma,\boldsymbol{e})]=\operatorname{Aut}_{\boldsymbol{e}^{\prime}\boldsymbol{e}}(\mathfrak{C})$
for $\boldsymbol{e}^{\prime}\in\operatorname{bdd}(\boldsymbol{e})$. Thus by
commutativity of the diagram in Remark 2.4,
$H=\xi[\operatorname{Aut}_{\boldsymbol{e}^{\prime}\boldsymbol{e}}(\mathfrak{C})]=\operatorname{Aut}_{\boldsymbol{e}^{\prime}\boldsymbol{e}}(\Sigma)$
and so $c\equiv^{H}d$ if and only if
$c\equiv_{\boldsymbol{e}^{\prime}\boldsymbol{e}}d$. By Fact 1.17(2),
$c\equiv^{\operatorname{KP}}_{\boldsymbol{e}}d$ if and only if
$c\equiv_{\operatorname{bdd}(\boldsymbol{e})}d$, and
$\boldsymbol{e}^{\prime}\in\operatorname{bdd}(\boldsymbol{e})$, thus
$c\equiv^{\operatorname{KP}}_{\boldsymbol{e}}d$ implies $c\equiv^{H}d$.

By Lemma 2.11,
$\operatorname{Gal}_{\operatorname{L}}^{c}(\Sigma,\boldsymbol{e})\leq\operatorname{Autf_{\operatorname{KP}}}(\Sigma,\boldsymbol{e})/\operatorname{Autf_{\operatorname{L}}}(\Sigma,\boldsymbol{e})$.
We already have proved that $c\equiv^{\operatorname{KP}}_{\boldsymbol{e}}d$
implies $c\equiv^{H}d$, and by Remark 2.8,
$c\equiv^{\operatorname{Autf_{\operatorname{KP}}}(\Sigma,\boldsymbol{e})}d$ if
and only if $c\equiv^{\operatorname{KP}}_{\boldsymbol{e}}d$, so it follows
that $c\equiv^{H}d$ if and only if
$c\equiv^{\operatorname{KP}}_{\boldsymbol{e}}d$.

(2): By Fact 1.17(3), if $c\equiv^{\operatorname{S}}_{\boldsymbol{e}}d$, then
$c\equiv_{\operatorname{acl}(\boldsymbol{e})}d$, so by Lemma 2.13,
$c\equiv^{H}d$. Conversely, assume $c\equiv^{H}d$. By Fact 1.17(3), it
suffices to show that if $F$ is an $\boldsymbol{e}$-invariant equivalence
relation and $c_{F}$ has finitely many automorphic images over
$\boldsymbol{e}$, then $F(c,d)$ holds.

Since $F$ is $\boldsymbol{e}$-invariant and
$\boldsymbol{e}\in\operatorname{dcl}(b)$, $F$ is type-definable over $b$. Then
by the same method of Fact 1.18, there is an $\emptyset$-type-definable
equivalence relation $F^{\prime}$ such that $c_{F}$ and $(cb)_{F^{\prime}}$
are interdefinable over $\boldsymbol{e}$. But then $(cb)_{F^{\prime}}$ is a
hyperimaginary in $\operatorname{acl}(\boldsymbol{e})$ and $cb$ is a tuple of
realizations of $\Sigma$, thus
$(cb)_{F^{\prime}}\in\operatorname{acl}(\boldsymbol{e})\cap\Sigma$. By Lemma
2.13, we know that there is
$f\in\operatorname{Aut}_{\operatorname{acl}(\boldsymbol{e})\cap\Sigma}(\mathfrak{C})$
such that $f(c)=d$. Then
$f\in\operatorname{Aut}_{(cb)_{F^{\prime}}}(\mathfrak{C})=\operatorname{Aut}_{c_{F}}(\mathfrak{C})$
and so $F(c,d)$. ∎

Now we can give an answer to one of the main questions of this section, which
says that the quotient groups of relativized KP and Shelah automorphisms are
exactly the canonical normal subgroups of the relativized Lascar group.

######  Theorem 2.15.

  1. (1)

$\operatorname{Gal}_{\operatorname{L}}^{c}(\Sigma,\boldsymbol{e})=\operatorname{Autf_{\operatorname{KP}}}(\Sigma,\boldsymbol{e})/\operatorname{Autf_{\operatorname{L}}}(\Sigma,\boldsymbol{e})$.

  2. (2)

$\operatorname{Gal}_{\operatorname{L}}^{0}(\Sigma,\boldsymbol{e})=\operatorname{Autf_{\operatorname{S}}}(\Sigma,\boldsymbol{e})/\operatorname{Autf_{\operatorname{L}}}(\Sigma,\boldsymbol{e})$.

###### Proof.

(1): Let
$H_{1}=\pi_{b}^{-1}[\operatorname{Gal}_{\operatorname{L}}^{c}(\Sigma,\boldsymbol{e})]$
and $H_{2}=\operatorname{Autf_{\operatorname{KP}}}(\Sigma,\boldsymbol{e})$. By
Remark 2.8 and Proposition 2.14, we have that $\equiv^{H_{1}}$ and
$\equiv^{H_{2}}$ are the same equivalence relations on the tuples of
realizations of $\Sigma(\mathfrak{C})$. So, by Lemma 2.9, $H_{1}=H_{2}$.

(2): By exactly the same proof as (1), letting
$H_{1}=\pi_{b}^{-1}[\operatorname{Gal}_{\operatorname{L}}^{0}(\Sigma,\boldsymbol{e})]$
and $H_{2}=\operatorname{Autf_{\operatorname{S}}}(\Sigma,\boldsymbol{e})$. ∎

We summarize the results obtained to prove Theorem 2.15 in this section, and
state the characterization of strong types in $\Sigma$, which is analogous to
Fact 1.17, in the following corollary. It says that having the same strong
type can be, in some sense, localized by finding a partial type that contains
the real tuples of interest.

######  Corollary 2.16.

Let $c$ and $d$ be tuples of realizations of $\Sigma$.

  1. (1)

The following are equivalent.

    1. (a)

$c\equiv^{\operatorname{KP}}_{\boldsymbol{e}}d$.

    2. (b)

$c\equiv_{\operatorname{bdd}(\boldsymbol{e})\cap\Sigma}d$.

    3. (c)

There is
$\sigma\in\pi_{b}^{-1}[\operatorname{Gal}_{\operatorname{L}}^{c}(\Sigma,\boldsymbol{e})]$
such that $\sigma(c)=d$.

  2. (2)

The following are equivalent.

    1. (a)

$c\equiv^{\operatorname{S}}_{\boldsymbol{e}}d$

    2. (b)

$c\equiv_{\operatorname{acl}(\boldsymbol{e})\cap\Sigma}d$.

    3. (c)

There is
$\sigma\in\pi_{b}^{-1}[\operatorname{Gal}_{\operatorname{L}}^{0}(\Sigma,\boldsymbol{e})]$
such that $\sigma(c)=d$.

###### Proof.

(1): $(a)\Rightarrow(b)$ is clear by Fact 1.17(2). By Lemma 2.13,
$\pi_{b}^{-1}[\operatorname{Gal}_{\operatorname{L}}^{c}(\Sigma,\boldsymbol{e})]=\operatorname{Aut}_{\operatorname{bdd}(\boldsymbol{e})\cap\Sigma}(\Sigma)$,
so $(b)$ and $(c)$ are equivalent. $(c)\Rightarrow(a)$ follows from
Proposition 2.14. The proof for (2) is similar. ∎

##  3\. Restricted model theoretic Galois groups

There is another possible approach to consider the subgroups of the
automorphism group in $\Sigma$, by restricting the domains of the
automorphisms in the subgroup.

######  Definition 3.1.

For $X\in\\{\operatorname{L},\operatorname{KP},\operatorname{S}\\}$,

$\operatorname{Autf}_{X}^{\operatorname{res}}(\Sigma,\boldsymbol{e})=\\{f\upharpoonright\Sigma(\mathfrak{C}):f\in\operatorname{Autf}_{X}(\mathfrak{C},\boldsymbol{e})\\}\leq\operatorname{Aut}_{\boldsymbol{e}}(\Sigma).$

######  Remark 3.2.

  1. (1)

For $X\in\\{\operatorname{L},\operatorname{KP},\operatorname{S}\\}$,
$\operatorname{Autf}_{X}^{\operatorname{res}}(\Sigma,\boldsymbol{e})\leq\operatorname{Autf}_{X}(\Sigma,\boldsymbol{e})$.

  2. (2)

Let
$H_{1}=\operatorname{Autf}_{X}^{\operatorname{res}}(\Sigma,\boldsymbol{e})$
and $H_{2}=\operatorname{Autf}_{X}(\Sigma,\boldsymbol{e})$. Then
$\equiv^{H_{1}}$ and $\equiv^{H_{2}}$ are the same.

###### Proof.

(1) is clear, and for (2), we only need to show that $c\equiv^{H_{2}}d$
implies $c\equiv^{H_{1}}d$ where $c,d$ are tuples of realizations of $\Sigma$.
By Remark 2.8, $\equiv^{H_{2}}$ is the same as $\equiv^{X}_{\boldsymbol{e}}$,
thus $c\equiv^{H_{2}}d$ implies $c\equiv^{X}_{\boldsymbol{e}}d$ and hence
there is $f\in\operatorname{Autf}_{X}(\mathfrak{C},\boldsymbol{e})$ such that
$f(c)=d$. So, $c\equiv^{H_{1}}d$. ∎

Of course, we can relativize or take restriction for any subgroup of the
automorphism group. We will use the following notation for general subgroups.

######  Definition 3.3.

Given a subgroup $H\leq\operatorname{Aut}_{\boldsymbol{e}}(\mathfrak{C})$, let
$\equiv^{H}$ be an orbit equivalence relation.

  1. (1)

$\operatorname{Aut}_{\equiv^{H}}(\Sigma,\boldsymbol{e})=\\{\sigma\in\operatorname{Aut}_{\boldsymbol{e}}(\Sigma):$
for any tuple $c$ of realizations of $\Sigma$, $c\equiv^{H}\sigma(c)\\}$.

  2. (2)

$H^{\operatorname{res}}=\\{f\upharpoonright\Sigma(\mathfrak{C}):f\in H\\}$.

######  Remark 3.4.

  1. (1)

For any $H\leq\operatorname{Aut}_{\boldsymbol{e}}(\mathfrak{C})$,
$H^{\operatorname{res}}\leq\operatorname{Aut}_{\equiv^{H}}(\Sigma,\boldsymbol{e})$.

  2. (2)

For $X\in\\{\operatorname{L},\operatorname{KP},\operatorname{S}\\}$,

    1. (a)

$\operatorname{Autf}_{X}(\Sigma,\boldsymbol{e})=\operatorname{Aut}_{\equiv^{X}_{\boldsymbol{e}}}(\Sigma,\boldsymbol{e})$,
and

    2. (b)

$\operatorname{Autf}_{X}^{\operatorname{res}}(\Sigma,\boldsymbol{e})=\operatorname{Autf}_{X}(\mathfrak{C},\boldsymbol{e})^{\operatorname{res}}$.

While it is unclear whether two groups $H^{\operatorname{res}}$ and
$\operatorname{Aut}_{\equiv^{H}}(\Sigma,\boldsymbol{e})$ are the same in
general, at least, we can ascertain the following fact.

######  Proposition 3.5.

Let $H\leq\operatorname{Aut}_{\boldsymbol{e}}(\mathfrak{C})$ such that
$\equiv^{H}$ is weaker than $\equiv^{\operatorname{L}}_{\boldsymbol{e}}$ (that
is, if $c\equiv^{\operatorname{L}}_{\boldsymbol{e}}d$, then $c\equiv^{H}d$).
Then

$H^{\operatorname{res}}\operatorname{Autf_{\operatorname{L}}}(\Sigma,\boldsymbol{e})=\operatorname{Aut}_{\equiv^{H}}(\Sigma,\boldsymbol{e})$

where
$H^{\operatorname{res}}\operatorname{Autf_{\operatorname{L}}}(\Sigma,\boldsymbol{e})=\\{g_{1}g_{2}\in\operatorname{Aut}_{\boldsymbol{e}}(\Sigma):g_{1}\in
H^{\operatorname{res}}$,
$g_{2}\in\operatorname{Autf_{\operatorname{L}}}(\Sigma,\boldsymbol{e})\\}$.
Especially, for
$X\in\\{\operatorname{L},\operatorname{KP},\operatorname{S}\\}$,
$\operatorname{Autf}_{X}^{\operatorname{res}}(\Sigma,\boldsymbol{e})\operatorname{Autf_{\operatorname{L}}}(\Sigma,\boldsymbol{e})=\operatorname{Autf}_{X}(\Sigma,\boldsymbol{e})$.

###### Proof.

$(\subseteq)$: Let $\sigma\in
H^{\operatorname{res}}\operatorname{Autf_{\operatorname{L}}}(\Sigma,\boldsymbol{e})$,
so that $\sigma=\tau\tau^{\prime}$ for some $\tau\in H^{\operatorname{res}}$
and
$\tau^{\prime}\in\operatorname{Autf_{\operatorname{L}}}(\Sigma,\boldsymbol{e})$.
It is enough to show that for a Lascar tuple $b$, $\sigma(b)\equiv^{H}b$, and

$\sigma(b)=\tau\tau^{\prime}(b)\equiv^{H}\tau^{\prime}(b)\equiv^{H}b.$

$(\supseteq)$: Take
$\sigma\in\operatorname{Aut}_{\equiv^{H}}(\Sigma,\boldsymbol{e})$. For a
Lascar tuple $b$ in $\Sigma$, $b\equiv^{H}\sigma(b)$, thus there is $\tau\in
H$ such that $\tau(b)=\sigma(b)$. Then letting
$\tau_{\Sigma}=\tau\upharpoonright\Sigma(\mathfrak{C})$, we have

$\tau_{\Sigma}^{-1}\sigma\in\operatorname{Aut}_{b}(\Sigma)\leq\operatorname{Autf_{\operatorname{L}}}(\Sigma,\boldsymbol{e}),$

and so

$\sigma\in\tau_{\Sigma}\cdot\operatorname{Autf_{\operatorname{L}}}(\Sigma,\boldsymbol{e})\subseteq
H^{\operatorname{res}}\operatorname{Autf_{\operatorname{L}}}(\Sigma,\boldsymbol{e}).$

∎

######  Example 3.6.

We consider a multi-sorted structure in [CLPZ01, Section 4]. Let
$\mathcal{M}=(M_{n},S_{n},g_{n},\pi_{n})_{n\geq 1}$, where for each $n\geq 1$,

  * •

$M_{n}$ is a unit circle,

  * •

$S_{n}$ is a ternary relation on $M_{n}$ saying that $S_{n}(a,b,c)$ holds if
and only if $a,b,c$ are distinct and $b$ comes before $c$ going around the
circle clockwise starting from $a$,

  * •

$g_{n}$ is a unary function on $M_{n}$ taking the clockwise rotation by
$\frac{2\pi}{n}$-radians,

  * •

$\pi_{n}$ is a function from $M_{n}$ to $M_{1}$ such that $x\mapsto x^{n}$.

Put $T:=\operatorname{Th}(\mathcal{M})$ and for each $n\geq 1$,
$T_{n}:=\operatorname{Th}(M_{n},S_{n},g_{n})$. Then, by [CLPZ01, Section 4]
(or [DKKL21, Section 3.1]), we have

  * •

$T$ and each $T_{n}$ have quantifier elimination,

  * •

$\operatorname{Gal}_{\operatorname{L}}(T)$ is not $G$-compact, that is,
$\operatorname{Gal}_{\operatorname{L}}(T)$ is not Hausdorff, and

  * •

each $\operatorname{Gal}_{\operatorname{L}}(T_{n})$ is trivial, so
$G$-compact.

Let $\mathfrak{C}=(\mathfrak{M}_{n},S_{n},g_{n},\pi_{n})_{n\geq 1}$ be a
monster model of $T$. Now let $\Sigma(x)$ be the sort corresponding to
$\mathfrak{M}_{1}$. Then, since $\operatorname{Gal}_{\operatorname{L}}(T_{1})$
is trivial, we have

$\operatorname{Aut}(\mathfrak{M_{1}})=\operatorname{Aut}(\Sigma)=\operatorname{Autf_{\operatorname{L}}}(\Sigma)=\operatorname{Autf_{\operatorname{L}}^{res}}(\Sigma).$

By Remark 3.2, we know that
$\operatorname{Autf}_{X}^{\operatorname{res}}(\Sigma,\boldsymbol{e})$ is dense
in $\operatorname{Autf}_{X}(\Sigma,\boldsymbol{e})$ after equipping the
pointwise convergence topology on both of
$\operatorname{Autf}_{X}(\Sigma,\boldsymbol{e})$ and
$\operatorname{Autf}_{X}^{\operatorname{res}}(\Sigma,\boldsymbol{e})$, that
is, defining basic open sets on
$\operatorname{Autf}_{X}(\Sigma,\boldsymbol{e})$ of the form
$\\{\sigma\in\operatorname{Autf}_{X}(\Sigma,\boldsymbol{e}):\sigma(c)=d\\}$
for finite tuples $c,d$ of realizations of $\Sigma$ and the same on
$\operatorname{Autf}_{X}^{\operatorname{res}}(\Sigma,\boldsymbol{e})$. We
close this paper with asking whether two groups
$\operatorname{Autf}_{X}(\Sigma,\boldsymbol{e})$ and
$\operatorname{Autf}_{X}^{\operatorname{res}}(\Sigma,\boldsymbol{e})$ coincide
or not.

######  Question 3.7.

For $X\in\\{\operatorname{L},\operatorname{KP},\operatorname{S}\\}$,
$\operatorname{Autf}_{X}^{\operatorname{res}}(\Sigma,\boldsymbol{e})=\operatorname{Autf}_{X}(\Sigma,\boldsymbol{e})$?

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