ON SOME FRAïSSé LIMITS WITH FREE AMALGAMATION

Yvon Bossut

Abstract : In the first section of this work we discuss some curiosities
around stable Kim-forking. In the second section we give a general way of
building some known and some new examples of NSOP1 theories as the limit of
some Fraïssé class satisfying some strong conditions. These limits will
satisfy existence, $\mathrel{\mathop{\vbox{
\hbox{\oalign{\kern-1.29167pt\cr\hfil$|$\hfil\cr\kern-3.01385pt\cr$\smile$\cr\kern-1.29167pt\cr}}
}}\displaylimits}^{K}=\mathrel{\mathop{\vbox{
\hbox{\oalign{\kern-1.29167pt\cr\hfil$|$\hfil\cr\kern-3.01385pt\cr$\smile$\cr\kern-1.29167pt\cr}}
}}\displaylimits}^{a}$ and $\mathrel{\mathop{\vbox{
\hbox{\oalign{\kern-1.29167pt\cr\hfil$|$\hfil\cr\kern-3.01385pt\cr$\smile$\cr\kern-1.29167pt\cr}}
}}\displaylimits}^{f}=\mathrel{\mathop{\vbox{
\hbox{\oalign{\kern-1.29167pt\cr\hfil$|$\hfil\cr\kern-3.01385pt\cr$\smile$\cr\kern-1.29167pt\cr}}
}}\displaylimits}^{M}$ over arbitrary sets, and they come with a stationary
independence relation. This study is based on the results of Baudish, Ramsey,
Chernikov and Kruckman.111Partially supported by ANR-DFG AAPG2019 GeoMod

##  1 About stable Kim-forking

###  1.1 In General

We work here in $T$ a complete theory, all sets of parameters and elements are
assumed to be in a monster model $\mathbb{M}$ of $T$. We will write
$\mathrel{\mathop{\vbox{
\hbox{\oalign{\kern-1.29167pt\cr\hfil$|$\hfil\cr\kern-3.01385pt\cr$\smile$\cr\kern-1.29167pt\cr}}
}}\displaylimits}^{K}$ for Kim-forking independence and
$\mathrel{\mathop{\vbox{
\hbox{\oalign{\kern-1.29167pt\cr\hfil$|$\hfil\cr\kern-3.01385pt\cr$\smile$\cr\kern-1.29167pt\cr}}
}}\displaylimits}^{f}$ for forking independence.

######  Definition 1.

Say a theory T has stable Kim-forking over models if whenever
$a\centernot\mathrel{\mathop{\vbox{
\hbox{\oalign{\kern-1.29167pt\cr\hfil$|$\hfil\cr\kern-3.01385pt\cr$\smile$\cr\kern-1.29167pt\cr}}
}}\displaylimits}^{K}_{M}N$ for $M\subseteq N$ models there is a stable
formula $\varphi(x,y)\in\mathcal{L}$ and $n\in N$ such that
$a\models\varphi(x,n)$ and $\varphi(x,n)$ Kim-forks over $M$. Same for stable
forking with $a\centernot\mathrel{\mathop{\vbox{
\hbox{\oalign{\kern-1.29167pt\cr\hfil$|$\hfil\cr\kern-3.01385pt\cr$\smile$\cr\kern-1.29167pt\cr}}
}}\displaylimits}^{f}_{M}N$.

######  Definition 2.

Say a theory T has weak stable Kim-forking over models if whenever
$a\centernot\mathrel{\mathop{\vbox{
\hbox{\oalign{\kern-1.29167pt\cr\hfil$|$\hfil\cr\kern-3.01385pt\cr$\smile$\cr\kern-1.29167pt\cr}}
}}\displaylimits}^{K}_{M}N$ for $M\subseteq N$ models there is a stable
formula $\varphi(x,y,m)\in\mathcal{L}_{m}$(i.e. the formula with fixed $m\in
M$ is stable) and $n\in N$ such that $a\models\varphi(x,n,m)$ and
$\varphi(x,n,m)$ Kim-forks over $M$. Same for weak stable forking with
$a\centernot\mathrel{\mathop{\vbox{
\hbox{\oalign{\kern-1.29167pt\cr\hfil$|$\hfil\cr\kern-3.01385pt\cr$\smile$\cr\kern-1.29167pt\cr}}
}}\displaylimits}^{f}_{M}N$.

It is known that a theory with stable forking is simple (see [5, Proposition
1.3]). We show that a theory with stable Kim-forking over models is simple and
that a theory with weak stable Kim-forking over models is NSOP1 and not
necessarily simple. We also show that a theory with weak stable forking over
models is simple.

######  Lemma 3.

Let $M\subseteq N$ be models of a theory $T$ with weak stable Kim-forking over
models. For any type $p=\mbox{\rm tp}(a/N)$ we have $a\mathrel{\mathop{\vbox{
\hbox{\oalign{\kern-1.29167pt\cr\hfil$|$\hfil\cr\kern-3.01385pt\cr$\smile$\cr\kern-1.29167pt\cr}}
}}\displaylimits}^{K}_{M}N$ if and only if for every stable formula
$\varphi(x,y,m)$ with $m\in M$ the canonical parameter
$Cb(d_{p}x\varphi(x,y,m))$ is in $acl^{eq}(M)$.

###### Proof.

$[\implies]$ If for some stable formula $\varphi(x,y,m)$ with $m\in M$ the
canonical parameter $Cb(d_{p}x\varphi(x,y,m))$ is not in the bounded closure
of the basis $M$, so in particular if $c$ is a representative then $c\notin M$
and we can take an infinite $M$ coheir Morley sequence in $\mbox{\rm
tp}(c/M)$, the type of $a$ over this sequence would then divide since its
$\varphi(x,y,m)$-type would have different definitions, so $p$ Kim-forks over
$M$. This way does not use the hypothesis. $[\impliedby]$ Conversely assume
that $p$ Kim-forks over $M$ and that $M$ contains the canonical parameters.
Then there is a stable formula $\varphi(x,y,m)$ with $m\in M$ and $n_{0}\in N$
such that $\varphi(x,n_{0},m)\in p$ and $\varphi(x,n_{0},m)$ Kim-forks over
$M$. So there is an $M$ coheir Morley sequence $(n_{i})_{i<\omega}$ such that
$\\{\varphi(x,n_{i},m)$ : $i<\omega\\}$ is inconsistent. Now since
$Cb(d_{p}x\varphi(x,y,m))\in acl^{eq}(M)$ by indiscernibility we have $\models
d_{p}x\varphi(x,n_{i},m)$ for all $i<\omega$, so $\\{\varphi(x,n_{i},m)$ :
$i<\omega\\}$ is satisfied by any realisation of a non-forking extension of
$p$ to $N(n_{i})_{i<\omega}$, a contradiction. ∎

In the same fashion and with a similar proof we have the following :

######  Lemma 4.

Let $M\subseteq N$ be models of a theory $T$ with weak stable forking over
models. For any type $p=\mbox{\rm tp}(a/N)$ we have $a\mathrel{\mathop{\vbox{
\hbox{\oalign{\kern-1.29167pt\cr\hfil$|$\hfil\cr\kern-3.01385pt\cr$\smile$\cr\kern-1.29167pt\cr}}
}}\displaylimits}^{f}_{M}N$ if and only if for every stable formula
$\varphi(x,y,m)$ with $m\in M$ the canonical parameter
$Cb(d_{p}x\varphi(x,y,m))$ is in $acl^{eq}(M)$.

######  Proposition 5.

Let $T$ be a theory with weak stable Kim-forking over models, then $T$ is
NSOP1.

###### Proof.

We use here the second point of [16, Theorem 3.9]. Assume that $T$ has weak
stable Kim-forking and that we have $(M_{i})_{i<\kappa}$ for $\kappa=|T|^{+}$
a continuous increasing sequence of models of size $|T|$ and a tuple $a$ such
that $a\centernot\mathrel{\mathop{\vbox{
\hbox{\oalign{\kern-1.29167pt\cr\hfil$|$\hfil\cr\kern-3.01385pt\cr$\smile$\cr\kern-1.29167pt\cr}}
}}\displaylimits}^{K}_{M_{i}}M_{i+1}$ for all $i<\kappa$. Write
$M=\bigcup_{i<\kappa}M_{i}$. For every $i<\kappa$ the number of stable
formulas with parameters in $M_{i}$ is at most $|T|$, so since $\kappa$ is
regular there is a smaller index $\alpha_{i}\geq i$ such that for every stable
formula $\varphi(x,y,e)$ with $e\in M_{i}$, if we write $p=\mbox{\rm
tp}(a/M)$, $Cb(d_{p}x\varphi(x,y,e))\in M_{\alpha_{i}}$. Now let us consider
$(\beta_{i})_{i<\omega}$ defined by induction by $\beta_{0}=0$ and
$\beta_{i+1}=\alpha_{\beta_{i}}$. Take $\beta=\bigcup_{i<\omega}\beta_{i}$, we
have that $\alpha_{\beta}=\beta$, so by Lemma 3 $a\mathrel{\mathop{\vbox{
\hbox{\oalign{\kern-1.29167pt\cr\hfil$|$\hfil\cr\kern-3.01385pt\cr$\smile$\cr\kern-1.29167pt\cr}}
}}\displaylimits}^{K}_{M_{\beta}}M$, which contradicts
$a\centernot\mathrel{\mathop{\vbox{
\hbox{\oalign{\kern-1.29167pt\cr\hfil$|$\hfil\cr\kern-3.01385pt\cr$\smile$\cr\kern-1.29167pt\cr}}
}}\displaylimits}^{K}_{M_{\beta}}M_{\beta+1}$.∎

######  Proposition 6.

Let $T$ be a theory with weak stable forking over models, then $T$ is simple.

###### Proof.

We show that $\mathrel{\mathop{\vbox{
\hbox{\oalign{\kern-1.29167pt\cr\hfil$|$\hfil\cr\kern-3.01385pt\cr$\smile$\cr\kern-1.29167pt\cr}}
}}\displaylimits}^{f}$ satisfies local character. Let $a$ be a finite tuple,
$M$ a model, $p=\mbox{\rm tp}(a/M)$. We consider the set
$C_{0}=\\{Cb(d_{p}x\varphi(x,y))$ : $\varphi(x,y)\in\mathcal{L}$ a stable
formula$\\}$. Now we take a model $M_{0}\subseteq M$ such that $|M_{0}|=|T|$
and $M_{0}$ contains a representative of every element of $C_{0}$. We iterate
the process taking $C_{i+1}=\\{Cb(d_{p}x\varphi(x,y,e))$ :
$\varphi(x,y,e)\in\mathcal{L}_{e}$ a stable formula with $e\in M_{i}\\}$ and
$M_{i}\subseteq M_{i+1}\subseteq M$ such that $|M_{i+1}|=|T|$ and $M_{i+1}$
contains a representative of every element of $C_{i+1}$. Now if we set
$M^{\prime}=\bigcup_{i<\omega}$ by Lemma 4 $a\mathrel{\mathop{\vbox{
\hbox{\oalign{\kern-1.29167pt\cr\hfil$|$\hfil\cr\kern-3.01385pt\cr$\smile$\cr\kern-1.29167pt\cr}}
}}\displaylimits}^{f}_{M^{\prime}}M$ and $M^{\prime}$ has size $|T|$. ∎

######  Corollary 7.

Let $T$ be a theory with stable Kim-forking over models, then $T$ is simple.

###### Proof.

By Proposition 5 we know that $T$ is NSOP1, now if $T$ is non-simple we can
build an arbitrary long sequence of Kim-forking extensions with an instance of
TP2 (as in [4, Proposition 3.6]). Now we can run the same proof as in [5,
Proposition 1.3] : we have a stable formula with a dividing chain of infinite
length and hence with the tree property, which is not possible.∎

###  1.2 In a theory with existence

We assume that the theory $T$ has existence, now we can define weak stable
Kim-forking over arbitrary sets in a similar fashion.

######  Definition 1.

Say a theory T with existence has weak stable Kim-forking if whenever
$a\centernot\mathrel{\mathop{\vbox{
\hbox{\oalign{\kern-1.29167pt\cr\hfil$|$\hfil\cr\kern-3.01385pt\cr$\smile$\cr\kern-1.29167pt\cr}}
}}\displaylimits}^{K}_{E}B$ for $E=acl(E)\subseteq B$ there is a stable
formula $\varphi(x,y,e)\in\mathcal{L}_{e}$(i.e. the formula with fixed $e\in
E$ is stable) and $b\in B$ such that $a\models\varphi(x,b,e)$ and
$\varphi(x,b,e)$ Kim-forks over $E$.

######  Lemma 2.

Let $E=acl(E)\subseteq B$ in a theory $T$ with weak stable Kim-forking. For
any type $p=\mbox{\rm tp}(a/B)$ we have $a\mathrel{\mathop{\vbox{
\hbox{\oalign{\kern-1.29167pt\cr\hfil$|$\hfil\cr\kern-3.01385pt\cr$\smile$\cr\kern-1.29167pt\cr}}
}}\displaylimits}^{K}_{E}B$ if and only if for every stable formula
$\varphi(x,y,e)$ with $e\in E$ the canonical parameter
$Cb(d_{p}x\varphi(x,y,e))$ is in $acl^{eq}(E)$.

###### Proof.

The proof is the same as the one of Lemma 3, we just replace coheir Morley
sequences by Morley sequences over $E$.∎

An example of such a theory is the theory of parameterized equivalence
relations (see [10, Section 6.3] and [3]). This theory has two sorts $O$ for
objects and $P$ for parameters, it has existence, and for arbitrary sets
$E\subseteq A,B$ we have that :

  1. 1.

$A\mathrel{\mathop{\vbox{
\hbox{\oalign{\kern-1.29167pt\cr\hfil$|$\hfil\cr\kern-3.01385pt\cr$\smile$\cr\kern-1.29167pt\cr}}
}}\displaylimits}^{K}_{E}B$ iff $A\cap B=E$ and $O(A)\mathrel{\mathop{\vbox{
\hbox{\oalign{\kern-1.29167pt\cr\hfil$|$\hfil\cr\kern-3.01385pt\cr$\smile$\cr\kern-1.29167pt\cr}}
}}\displaylimits}^{p}_{O(E)}O(B)$ for every $p\in P(E)$ where
$\mathrel{\mathop{\vbox{
\hbox{\oalign{\kern-1.29167pt\cr\hfil$|$\hfil\cr\kern-3.01385pt\cr$\smile$\cr\kern-1.29167pt\cr}}
}}\displaylimits}^{p}$ is forking independence in the theory $T_{\infty}$ of
an equivalence relation with an infinite number of classes all infinite.

  2. 2.

$A\mathrel{\mathop{\vbox{
\hbox{\oalign{\kern-1.29167pt\cr\hfil$|$\hfil\cr\kern-3.01385pt\cr$\smile$\cr\kern-1.29167pt\cr}}
}}\displaylimits}^{f}_{E}B$ iff $A\cap B=E$ and $O(A)\mathrel{\mathop{\vbox{
\hbox{\oalign{\kern-1.29167pt\cr\hfil$|$\hfil\cr\kern-3.01385pt\cr$\smile$\cr\kern-1.29167pt\cr}}
}}\displaylimits}^{p}_{O(E)}O(B)$ for every $p\in P(B)$.

The first condition is witnessed by stable formulas (namely equalities). For
the second one, the difference between Kim-forking and forking here is that in
the first case the parameter in inside of the basis. For a given parameter $p$
the formula $\varphi(x,y)=(x\equiv_{p}y)$ is stable since it defines an
equivalence relation, so Kim-forking is always witnessed by stable formulas in
this theory. Now if we do not fix $p$, the formula
$\psi(x,yz):=(x\equiv_{z}y)$ is unstable, so forking in not always witnessed
by stable formulas in this theory (which we already knew from Proposition 6).

The theory of vector spaces of infinite dimension over a field with weak
stable Kim-forking (so in particular over a stable field) and the model
companion of the empty theory with a symbol for a binary function (see [18]
the characterisation of Kim-forking in this theory) are other examples. A
difference with the previous example is that in these two cases there is a
non-trivial algebraic closure, and that Kim-forking independence is computed
for algebraically closed structures $E=acl(E)\subseteq A=acl(A),B=acl(B)$ and
is clearly witnessed by stable formulas in this case, so one could think that
something different might happen if we consider arbitrary sets. It is not the
case thanks to the two following facts, first in an NSOP1 theory with
existence $A\mathrel{\mathop{\vbox{
\hbox{\oalign{\kern-1.29167pt\cr\hfil$|$\hfil\cr\kern-3.01385pt\cr$\smile$\cr\kern-1.29167pt\cr}}
}}\displaylimits}^{K}_{E}B$ if and only if $acl(EA)\mathrel{\mathop{\vbox{
\hbox{\oalign{\kern-1.29167pt\cr\hfil$|$\hfil\cr\kern-3.01385pt\cr$\smile$\cr\kern-1.29167pt\cr}}
}}\displaylimits}^{K}_{acl(E)}acl(EB)$ for every sets $E,A,B$, and secondly :

######  Remark 3.

[5, Remark 1.2] Let $\varphi(x,y,e)$ be a stable formula (with some parameters
$e$). If $\theta(v,x,e)\vdash\exists^{=n}x\theta(v,x,e)$ then
$\psi(v,y,e)=\exists x(\theta(v,x,e)\wedge\varphi(x,y,e))$ is also stable.

This means that we can get a stable formula from another one by replacing our
variables by something ’bigger’ in the sense of the algebraic closure : Assume
that $a\centernot\mathrel{\mathop{\vbox{
\hbox{\oalign{\kern-1.29167pt\cr\hfil$|$\hfil\cr\kern-3.01385pt\cr$\smile$\cr\kern-1.29167pt\cr}}
}}\displaylimits}^{K}_{acl(E)}b$, then
$acl(Ea)\centernot\mathrel{\mathop{\vbox{
\hbox{\oalign{\kern-1.29167pt\cr\hfil$|$\hfil\cr\kern-3.01385pt\cr$\smile$\cr\kern-1.29167pt\cr}}
}}\displaylimits}^{K}_{acl(E)}acl(Eb)$, so there is a stable formula
$\varphi(x,y,e)$ with $e\in acl(E)$ and $\overline{a}\in acl(Ea)$,
$\overline{b}\in acl(Eb)$ such that
$\models\varphi(\overline{a},\overline{b},e)$ and $\varphi(x,\overline{b},e)$
Kim-divides over $E$. We can assume that $\overline{a}\in acl(ea)$,
$\overline{b}\in acl(eb)$. Then using Remark 3 we can find a stable formula
$\psi(x^{\prime},y^{\prime},e)$ such that $\models\psi(a,b,e)$ and
$\psi(x,b,e)$ Kim-divides over $acl(E)$ (for this last condition we take a
formula $\theta$ isolating the type of $\overline{b}$ over $acl(E)b$).

######  Remark 4.

If $T$ is an NSOP1 theories with existence, weak stable Kim-forking and
elimination of hyperimaginaries then $T^{eq}$ has weak canonical bases in the
sense of [17, Definition 4.1]. More generally, without the last assumption, if
$T$ is an NSOP1 theory with existence, weak stable Kim-forking and elimination
of imaginaries, then given some tuples $E=acl^{eq}(E)\subseteq B=acl^{eq}(B)$
and $a$ there is a smaller subset $E\subseteq B_{0}=acl^{eq}(B_{0})\subseteq
B$ such that $a\mathrel{\mathop{\vbox{
\hbox{\oalign{\kern-1.29167pt\cr\hfil$|$\hfil\cr\kern-3.01385pt\cr$\smile$\cr\kern-1.29167pt\cr}}
}}\displaylimits}^{K}_{B_{0}}B$, in the sense that given any $E\subseteq
B^{\prime}=acl^{eq}(B^{\prime})\subseteq B$ such that
$a\mathrel{\mathop{\vbox{
\hbox{\oalign{\kern-1.29167pt\cr\hfil$|$\hfil\cr\kern-3.01385pt\cr$\smile$\cr\kern-1.29167pt\cr}}
}}\displaylimits}^{K}_{B^{\prime}}B$ we have that $B_{0}\subseteq B^{\prime}$.

###  1.3 Imaginaries :

We adapt some results of the section $2$ of [5] where it is shown that if a
theory $T$ has stable forking so does $T^{eq}$ to our context, but we cannot
get a similar result (at least for the moment).

######  Lemma 1.

Let $T=T^{eq}$ be a theory with existence such that whenever
$a\centernot\mathrel{\mathop{\vbox{
\hbox{\oalign{\kern-1.29167pt\cr\hfil$|$\hfil\cr\kern-3.01385pt\cr$\smile$\cr\kern-1.29167pt\cr}}
}}\displaylimits}^{K}_{E}N$ for $N$ a model and $E\subseteq N$ there is $e\in
E$, $n\in N$, $\varphi(x,y,e)$ a stable formula such that
$\models\varphi(a,n,e)$ and $\varphi(x,n,e)$ Kim-forks over $E$. Then $T$ has
weak stable Kim-forking.

###### Proof.

Let $a$, $C\subseteq B$ such that $a\centernot\mathrel{\mathop{\vbox{
\hbox{\oalign{\kern-1.29167pt\cr\hfil$|$\hfil\cr\kern-3.01385pt\cr$\smile$\cr\kern-1.29167pt\cr}}
}}\displaylimits}^{K}_{E}B$. Let $N$ be a model containing $B$ such that
$a\mathrel{\mathop{\vbox{
\hbox{\oalign{\kern-1.29167pt\cr\hfil$|$\hfil\cr\kern-3.01385pt\cr$\smile$\cr\kern-1.29167pt\cr}}
}}\displaylimits}^{f}_{B}N$. So there is a stable formula $\psi(x,y,e)$ such
that $\psi(x,n,e)$ Kim-forks over $E$ for some $n\in N$. Then
$b^{\prime}=Cb(d_{p}x\psi(x,y,e))\in acl^{eq}(B)\setminus acl^{eq}(E)$. Let us
write $\chi(y,b,e)=d_{p}x\psi(x,y,e)$, then this formula is stable since it is
equivalent to a generalised $\psi(x,y,e)^{-1}$ formula. Now there is $b\in B$
and $\theta(v,w,e)$ such that $\theta(v,w,e)\vdash\exists^{=n}w\theta(v,w,e)$
and $\models\theta(b,b^{\prime},e)$. Then $\varphi(y,v,e)=\exists
w(\theta(v,w,e)\wedge\chi(y,w,e))$ is also stable by Remark 3. Now the formula
$\forall y(\psi(x,y,e)\leftrightarrow\varphi(y,v,e))$ is stable,
$\models\forall y(\psi(a,y,e)\leftrightarrow\varphi(a,b,e))$, and $\forall
y(\psi(x,y,e)\leftrightarrow\varphi(x,b,e))$ Kim-forks over $E$.∎

######  Proposition 2.

[5, Proposition 2.2] If $T$ has weak stable Kim-forking and existence, then
$T^{eq}$ has weak stable Kim-forking over real parameters, meaning that
whenever $a_{F}\centernot\mathrel{\mathop{\vbox{
\hbox{\oalign{\kern-1.29167pt\cr\hfil$|$\hfil\cr\kern-3.01385pt\cr$\smile$\cr\kern-1.29167pt\cr}}
}}\displaylimits}^{K}_{E}B$ for $E\subseteq B$ sets of real elements and
$a_{F}\in\mathbb{M}^{eq}$ the $F$ class of $a$ there is a stable formula
$\varphi(x,y,e)\in\mathcal{L}_{e}$ for $e\in E$ and $b\in B$ such that
$a\models\varphi(x,b,e)$ and $\varphi(x,b,e)$ Kim-forks over $E$.

###### Proof.

Let $a,E,B$ be as in the statement. We choose a saturated model $M\supseteq B$
such that $a_{F}\in dcl^{eq}(M)$ and there is a representative $a$ of $a_{F}$
such that $a\mathrel{\mathop{\vbox{
\hbox{\oalign{\kern-1.29167pt\cr\hfil$|$\hfil\cr\kern-3.01385pt\cr$\smile$\cr\kern-1.29167pt\cr}}
}}\displaylimits}^{K}_{a_{F}}M$ (using existence). Then
$a\centernot\mathrel{\mathop{\vbox{
\hbox{\oalign{\kern-1.29167pt\cr\hfil$|$\hfil\cr\kern-3.01385pt\cr$\smile$\cr\kern-1.29167pt\cr}}
}}\displaylimits}^{K}_{E}B$. By assumption there is a stable formula
$\varphi(x,y,e)$ for $e\in E$, and a tuple $b\in B$ such that
$\models\varphi(a,b,e)$ and $\varphi(x,b,e)$ Kim-forks over $E$. Write $p$ the
$\varphi(x,y,e)$-type of $a$ over $B$. Then $p$ has a definition
$d_{p}x\varphi(x,y,e)$ which is an instance of a stable formula. $p$ does not
Kim-fork over $a_{F}$ so its canonical basis $d_{H}$ is in $acl^{eq}(a_{F})$.
For some $\chi(w,y,e)\in\mathcal{L}^{eq}$, $\chi(d_{H},y)$ defines $p$. Since
$d_{p}x\varphi(x,y,e)\equiv\chi(d_{H},y)$ there is some $\mu(x)\in
tp(d_{F}/E)$ such that the formula $\chi(w,y)\wedge\mu(w)$ is stable. W.l.o.g.
w.c.a. that $\chi(w,y)$ is stable. Claim : If $q(w)=\mbox{\rm tp}(d_{H}/E)$,
then Kim-forks over $E$.

###### Proof.

Assume not, now if we consider an $E$ Morley sequence $(b_{i})_{i<\omega}$ in
$\mbox{\rm tp}(b/E)$ inside of $M$. We have that $q(w)\cup\\{\chi(w,b_{i})$ :
$i<\omega\\{$ is consistent, and hence there is a realisation $d^{\prime}_{H}$
of this type. $d_{H}\equiv_{E}d^{\prime}_{H}$ so there is a sequence
$(b_{i})_{i<\omega}$ in $M$ such that
$d_{H}(b^{\prime}_{i})_{i<\omega}\equiv_{E}d^{\prime}_{H}(b_{i})_{i<\omega}$.
Then $\models\chi(b_{i},d_{H})$ for all $i<\omega$, so $\varphi(x,b_{i},e)\in
p$ and $a\models\\{\varphi(x,b_{i},e)$ : $i<\omega\\}$, which contradicts that
$\varphi(x,b,e)$ Kim-forks over $E$, so $\\{\varphi(x,b_{i},e)$ :
$i<\omega\\}$ is $k$-inconsistent for some $k<\omega$.∎

With the claim we can choose $\mu(w)\in q(w)$ such that
$\chi(w,b)\wedge\mu(w)$ Kim-forks over $E$. Then
$\chi^{\prime}(w,y)=\chi(w,b)\wedge\mu(w)$ is stable and Kim-divides over $E$,
witnessed by an $E$-Morley sequence $(b{i})_{i<\omega}$ in $\mbox{\rm
tp}(b/E)$. Since $d_{H}\in acl^{eq}(a_{F})$ there is a formula
$\theta(v,w)\in\mathcal{L}^{eq}$ and an $n<\omega$ such that
$\models\theta(a_{F},d_{H})$ and $\theta(v,w)\vdash\exists^{=n}w\theta(v,w)$.
Let $\psi(v,y)=\exists w(\theta(v,w)\wedge\chi^{\prime}(w,y))$. Then this
formula is stable (by Remark 3 and the sequence $(b_{i})_{i<\omega}$ witnesses
that it Kim-divides over $E$. ∎

Our problem now is that given some set of imaginaries $E$ and some set of
representatives $E^{\prime}$ we cannot move from a stable formula
$\varphi(x,y,e^{\prime})$ with real parameters $e^{\prime}\in E^{\prime}$ to a
stable formula $\varphi(x,y,e^{\prime}_{R})$ with $R$ a definable equivalence
relation such that $e^{\prime}_{R}\in acl^{eq}(E)$ in an obvious way.

Some questions about this :

  1. 1.

Does existence and weak stable Kim-forking over models imply weak stable Kim-
forking over arbitrary sets?

  2. 2.

[5, Proposition 2.3] : If $T^{eq}$ has weak stable Kim-forking over real
parameters and existence, then does $T^{eq}$ has weak stable Kim-forking?

  3. 3.

More specifically, can we prove that if a simple theory $T$ has weak stable
forking then $T^{eq}$ also have weak stable forking?

  4. 4.

Is there an NSOP1 theory that does not have weak stable Kim-forking? Either
over models or algebraically closed sets if it has existence.

##  2 Adding generic structure to a Fraïssé class with free amalgamation

###  2.1 Properties of the limit

The bases of this study are [10, Section 6.3] and [2]. Baudisch’s construction
works in a way more general context, but what we focus on here is the study of
independence relations, which appears in his paper under some stronger
hypothesis (namely in [2, Theorem 4.1]). In this section we consider a
countable language $L$ and a class $K$ of countable $L$-structures with the
following properties :

  1. 1.

(H.P.) : If $A\subseteq B\in K$ is a substructure then $A\in K$.

  2. 2.

(Free Amalgamation) : If $C\subseteq A,B$ are structures in $K$ (where we
allow $C=\emptyset$ when the language does not have any constants) there is
$D\in K$ and some embeddings $i_{A},i_{B}$ of $A$ and $B$ respectively to $D$
satisfying $i_{A}|_{C}=i_{B}|_{C}$, $i_{A}(A)\cap i_{B}(B)=i_{A}(C)$, $D$ is
generated by the images of $A$ and $B$ and such that given any morphism of
$L$-structures $\varphi_{A}:A\rightarrow D^{\prime}$,
$\varphi_{B}:B\rightarrow D^{\prime}$ to $D^{\prime}\in K$ there is a unique
morphism $\varphi:D\rightarrow D^{\prime}$ such that $\varphi_{A}=\varphi\cdot
i_{A}$ and $\varphi_{B}=\varphi\cdot i_{B}$. We will write $A\oplus_{C}B$ for
this structure $D$.

  3. 3.

(Generic extension) : If $A\in K$ and $V$ is a sort there is an extension
$A\oplus_{\emptyset}\langle x\rangle\in K$ generated by a single element $x\in
V$ over $A$ such that for every morphism $\varphi:A\rightarrow A^{\prime}\in
K$ and every element $a^{\prime}\in V(A^{\prime})$ there is a unique morphism
$\psi:A\oplus_{\emptyset}\langle x\rangle\rightarrow A^{\prime}$ extending
$\varphi$ by sending $x$ to $a^{\prime}$.

  4. 4.

$K$ is axiomatisable : if the isomorphism class of an $L$-structure $A$ is not
inside of $K$ there is a quantifier free formula $\varphi$ and a finite tuple
$a\in A$ such that $A\models\varphi(a)$ and
$A^{\prime}\centernot\models\varphi(a^{\prime})$ for every $a^{\prime}\in
A^{\prime}\in K$.

  5. 5.

Any limit of $K$ has Q.E. : for every quantifier free formula $\varphi(x,y)$
satisfied in some structure of $K$ there is a quantifier free formula
$\psi(x)$ such that if $a\in A\in K$ and $A\models\psi(a)$ then there is
$A\subseteq B\in K$ an extension and $b\in B$ such that
$B\models\varphi(a,b)$.

  6. 6.

($3$ Amalgamation for algebraically independent extension) : Given a
commutative diagram in $K$ as on the left, where the arrows are embeddings,
$\varphi_{B_{i}}(B_{i})\cap\varphi_{A_{i}}(A)=\varphi_{A_{i}}\cdot
i_{A}(E)=\varphi_{B_{i}}\cdot i_{B_{i}}(E)$ for $i=0,1$ and
$\psi_{B_{0}}(B_{0})\cap\psi_{B_{1}}(B_{1})=\psi_{B_{0}}\cdot
i_{B_{0}}(E)=\psi_{B_{1}}\cdot i_{B_{1}}(E)$, we can complete it inside of $K$
into a commutative diagram as on the right (where the arrows are embeddings)
such that
$\varphi_{D_{i}}(D_{i})\cap\varphi_{B}(B)=\varphi_{D_{i}}\cdot\varphi_{B_{i}}(B_{i})=\varphi_{B}\cdot\psi_{B_{i}}(B_{i})$
for $i=0,1$ and
$\varphi_{D_{0}}(D_{0})\cap\varphi_{D_{1}}(D_{1})=\varphi_{D_{0}}\cdot\varphi_{A_{0}}(A)=\varphi_{D_{1}}\cdot\varphi_{A_{1}}(A)$
(the blank arrows of the right diagram are those of the left diagram, i did
not write the names for clarity).

${{D_{1}}}$${{D_{1}}}$${D}$${A}$${{D_{0}}}$${A}$${{D_{0}}}$${{B_{1}}}$${B}$${{B_{1}}}$${B}$${E}$${{B_{0}}}$${E}$${{B_{0}}}$$\scriptstyle{i_{B_{0}}}$$\scriptstyle{\psi_{B_{0}}}$$\scriptstyle{i_{B_{1}}}$$\scriptstyle{i_{A}}$$\scriptstyle{\varphi_{A_{0}}}$$\scriptstyle{\varphi_{B_{0}}}$$\scriptstyle{\varphi_{A_{1}}}$$\scriptstyle{\psi_{B_{1}}}$$\scriptstyle{\varphi_{D_{0}}}$$\scriptstyle{\varphi_{B}}$$\scriptstyle{\varphi_{D_{1}}}$

Notice that we do not assume (E.C.), i.e. that there are countably many
isomorphism class, nor that the structures in $K$ are finitely generated, also
to check the condition $5$ for a formula it is enough to check it for a
strengthening of this formula that is also realised in $K$. $6$ is a
strengthening of $3$-amalgamation for $\mathrel{\mathop{\vbox{
\hbox{\oalign{\kern-1.29167pt\cr\hfil$|$\hfil\cr\kern-3.01385pt\cr$\smile$\cr\kern-1.29167pt\cr}}
}}\displaylimits}^{a}$, in the usual $3$ amalgamation we would ask that
$D_{i}$ or $B$ are generated by the image of the two embeddings from $A$ and
$B_{i}$ or from $B_{0}$ and $B_{1}$ respectively, here however we allow
ourselves to take an extension of the generated structure. We will refer to
the conditions $1-6$ as $(H)$.

We begin by constructing a limit of $K$ and presenting its properties. We
assume the continuum hypothesis for the reason that we need an uncountable
cardinal $\kappa$ such that $2^{<\kappa}=\kappa$ and that it is convenient to
take $\aleph_{1}$ as such.

######  Definition 1.

Let $K$ be a class of countable structures in some language $L$. An $L$
structure $\mathcal{M}$ is $K$-$\aleph_{1}$-saturated if the collection of its
countable substructures correspond up to isomorphism to $K$ and if for every
$\mathcal{U},\mathcal{B}\in K$ countable and every embeddings
$f_{0}:\mathcal{U}\rightarrow\mathcal{M}$,
$f_{1}:\mathcal{U}\rightarrow\mathcal{B}$ there is some embedding
$g:\mathcal{B}\rightarrow\mathcal{M}$ such that $g\cdot f_{1}=f_{0}$.

######  Proposition 2.

There exist an $L$ structure $\mathcal{M}$ which is $K$-$\aleph_{1}$-saturated
and has cardinal $\aleph_{1}$. When $L$ does not have any constants every
structure of age included in $K$ and of cardinal $\leq\aleph_{1}$ embeds into
$\mathcal{M}$, and $\mathcal{M}$ is unique up to isomorphism. Otherwise the
age of $\mathcal{M}$ is the set of structures in $K$ that have the same
isomorphism type for the structure generated by the constants, and any such
structure of cardinal $\leq\aleph_{1}$ embeds in $\mathcal{M}$.

###### Proof.

Let $(\mathcal{B}_{i})_{i<\aleph_{1}}$ enumerate the elements of $K$ up to
isomorphism. We construct by induction an chain
$(\mathcal{U}_{i})_{i<\aleph_{1}}$ of $L_{P}$-structures in $K$. For every
$i,k<\aleph_{1}$ let $(f^{\mathcal{U}_{i},k}_{j})_{j<\aleph_{1}}$ enumerate
the embeddings of $\mathcal{U}_{i}$ in $\mathcal{B}_{k}$ (eventually if these
structure do not agree on the constants there are none). Let
$\mathcal{U}_{0}=\mathcal{B}_{0}$. Assume that $\mathcal{U}_{i}$ is defined
for every $i<\lambda$. By (Free Amalgamation) there is a structure
$\mathcal{U}^{\prime}_{\lambda}$ extending both
$\bigcup\limits_{i<\lambda}\mathcal{U}_{i}$ and $\mathcal{B}_{\lambda}$. We
define a chain $(\mathcal{U}^{k})_{k<\lambda}$ of $L_{P}$-structures extending
$\mathcal{U}^{\prime}_{\lambda}$, let
$\mathcal{U}^{0}=\mathcal{U}^{\prime}_{\lambda}$. Assume that every
$\mathcal{U}^{k}$ is defined for every $k<\mu$. We define $\mathcal{U}^{\mu}$
as the union of a chain $(\mathcal{U}^{\mu}_{i})_{i<\lambda}$ of
$L_{P}$-structures, let
$\mathcal{U}^{\mu}_{0}=\bigcup\limits_{i<\lambda,k<\mu}\mathcal{U}_{i}^{k}$.
Assume $\mathcal{U}^{\mu}_{i}$ is defined for every $i<\nu$, we consider the
embeddings
$f^{\mathcal{U}_{\mu},\nu}_{j}:\mathcal{U}_{\mu}\rightarrow\mathcal{B}_{\nu}$
for $j<\lambda$. $\mathcal{U}^{\mu}_{\nu}$ is constructed as the union of yet
another continuous chain $(\mathcal{U}^{\mu}_{\nu,j})_{j<\lambda}$.
$\mathcal{U}^{\mu}_{\nu,0}:=\bigcup\limits_{k<\mu}\mathcal{U}^{\mu}_{k}$, if
$\mathcal{U}^{\mu}_{\nu,j}$ is defined for every $j<\epsilon$ let
$\mathcal{U}^{\prime\mu}_{\nu,\epsilon}=\bigcup\limits_{j<\epsilon}\mathcal{U}^{\mu}_{\nu,j}$
and write $f=f^{\mathcal{U}_{\mu},\nu}_{\epsilon}$ by $S.A.P.$ there is a
structure $\mathcal{U}^{\mu}_{\nu,\epsilon}\in K$ extending
$\mathcal{U}^{\prime\mu}_{\nu,\epsilon}$ and an embedding
$g=g^{\mathcal{U}_{\mu},\nu}_{\epsilon}:\mathcal{B}_{\nu}\rightarrow\mathcal{U}^{\mu}_{\nu,\epsilon}$
such that $g\cdot
f:\mathcal{U}_{\mu}\rightarrow\mathcal{U}^{\mu}_{\nu,\epsilon}$ is the
inclusion. We now set
$\mathcal{U}_{\lambda}=\bigcup\limits_{i<\lambda}\mathcal{U}^{i}$ and
$\mathcal{M}=\bigcup\limits_{i<\aleph_{1}}\mathcal{U}_{i}$.

${{\mathcal{U}^{\prime\mu}_{\nu,\epsilon}}}$${{\mathcal{U}^{\mu}_{\nu,\epsilon}}}$${{\mathcal{U}_{\mu}}}$${{\mathcal{B}_{\nu}}}$$\scriptstyle{\subseteq}$$\scriptstyle{f}$$\scriptstyle{\subseteq}$$\scriptstyle{g}$

We now show that $\mathcal{M}$ is $K$-$\aleph_{1}$-saturated and has $age$
inside of $K$. Let $\mathcal{U}$ be a countable structure of $\mathcal{M}$,
$\mathcal{U}\subseteq\mathcal{U}_{i}$ for some $i<\aleph_{1}$ so by $(H.P.)$
$\mathcal{U}\in K$. Now lets consider an embedding
$f:\mathcal{U}\rightarrow\mathcal{B}\in K$. $\mathcal{B}$ is isomorphic to
$\mathcal{B}_{k}$ for some $k<\aleph_{1}$ via
$\varphi:\mathcal{B}\rightarrow\mathcal{B}_{k}$. By (Free Amalgamation) we can
find some $k^{\prime}<\aleph_{1}$ and some embeddings
$\psi:\mathcal{B}_{k}\rightarrow\mathcal{B}_{k^{\prime}}$ and
$f^{\prime}:\mathcal{U}_{i}\rightarrow\mathcal{B}_{k^{\prime}}$ such that
$\psi\cdot\varphi\cdot f=f^{\prime}$. Now
$f^{\prime}:\mathcal{U}_{i}\rightarrow\mathcal{B}_{k^{\prime}}$ is equal to
$f^{\mathcal{U}_{i},k^{\prime}}_{l}$ for some $l<\aleph_{1}$, and if we choose
a large enough $j$, by construction, there is a
$h=g^{\mathcal{U}_{i},k^{\prime}}_{l}:\mathcal{B}_{k^{\prime}}\rightarrow\mathcal{M}$
such that $h\cdot f^{\prime}$ is the inclusion
$\mathcal{U}_{i}\subseteq\mathcal{M}$. Then $h\cdot\psi\cdot\varphi$ is the
embedding we want.

${{\mathcal{U}_{j}}}$${{\mathcal{M}}}$${{\mathcal{U}_{i}}}$${{\mathcal{B}_{k^{\prime}}}}$${{\mathcal{U}_{\mu}}}$${{\mathcal{B}_{\nu}}}$${\bullet}$$\scriptstyle{f}$$\scriptstyle{\varphi}$$\scriptstyle{\subseteq}$$\scriptstyle{f^{\prime}}$$\scriptstyle{g}$$\scriptstyle{\subseteq}$$\scriptstyle{\subseteq}$$\scriptstyle{h}$

For the second point we proceed like as in Fraïssé theory and only use the
property of $K$-saturation and the fact that we can amalgamate over either the
empty set or the structure generated by the constants : if
$\mathcal{M}^{\prime}$ is a structure of age inside of $K$ and of cardinal
$\leq\aleph_{1}$ we fix an enumeration $(m_{i})_{i<\aleph_{1}}$ and we define
by induction an increasing sequence of embeddings $f_{i}:\langle
m_{<i}\rangle\rightarrow\mathcal{M}$, and the union
$f=\bigcup\limits_{i<\aleph_{1}}f_{i}:\mathcal{M}\rightarrow\mathcal{M}^{\prime}$
is our embedding. We need to amalgamate over the empty set for the first step
of our induction : take some $a\in\mathcal{M}$, and consider $\langle
a\rangle\subseteq\mathcal{M}$, we can embed $\langle a\rangle$ into $\langle
a\rangle\oplus_{\emptyset}\langle m_{0}\rangle$, by saturation we can find
some substructure $\langle m^{\prime}_{0}\rangle\subseteq\mathcal{M}$
isomorphic to $\langle m_{0}\rangle$ and we take $f_{0}$ to be this
isomorphism. In a similar fashion, to show uniqueness we construct an
isomorphism by back and forth.∎

######  Definition 3.

Let $T$ be the theory of $\mathcal{M}$ and $\mathbb{M}$ be a monster model of
$T$. When $L$ does not have constants this theory is entirely determined by
the data of $K$ by the last part of Proposition 2, and we will refer to $T$ as
the theory of the limit of $K$. Otherwise we will refer to the theory of
$\mathcal{M}$ as a limit of $K$, in that case each limit correspond to an
isomorphism type of the structure generated by the constants.

######  Remark 4.

J.E.P. holds in $K$ if and only if all the structures of $K$ agree on the
structure generated by the constants.

######  Remark 5.

A consequence of the (Free Amalgamation) property is that
$dcl(A)=acl(A)=\langle A\rangle$ for every small set $A\subseteq\mathbb{M}$.

######  Lemma 6.

$T$ has Q.E. in $L$.

###### Proof.

This is clear from the condition $5$, if we consider any
$\aleph_{1}$-saturated model it is $K$-$\aleph_{1}$-saturated, so partial
isomorphism between finitely generated structures have the back and forth
property. ∎

######  Lemma 7.

The relation $\mathrel{\mathop{\vbox{
\hbox{\oalign{\kern-1.29167pt\cr\hfil$|$\hfil\cr\kern-3.01385pt\cr$\smile$\cr\kern-1.29167pt\cr}}
}}\displaylimits}^{a}$ defined on subsets of $\mathbb{M}$ the monster model of
a limit by $A\mathrel{\mathop{\vbox{
\hbox{\oalign{\kern-1.29167pt\cr\hfil$|$\hfil\cr\kern-3.01385pt\cr$\smile$\cr\kern-1.29167pt\cr}}
}}\displaylimits}^{a}_{E}B$ if $acl(EA)\cap acl(EB)=acl(E)$ satisfies the
independence theorem, i.e. that if $E,A_{i},B_{i}\subseteq\mathbb{M}$ for
$i\in\\{0,1\\}$, $A_{0}\equiv_{E}A_{1}$, $A_{i}\mathrel{\mathop{\vbox{
\hbox{\oalign{\kern-1.29167pt\cr\hfil$|$\hfil\cr\kern-3.01385pt\cr$\smile$\cr\kern-1.29167pt\cr}}
}}\displaylimits}^{a}_{E}B_{i}$ for every $i\in\\{0,1\\}$ and
$B_{0}\mathrel{\mathop{\vbox{
\hbox{\oalign{\kern-1.29167pt\cr\hfil$|$\hfil\cr\kern-3.01385pt\cr$\smile$\cr\kern-1.29167pt\cr}}
}}\displaylimits}^{a}_{E}B_{1}$ there is an $A$ such that
$A\equiv_{B_{i}}A_{i}$ for every $i\in\\{0,1\\}$ and $A\mathrel{\mathop{\vbox{
\hbox{\oalign{\kern-1.29167pt\cr\hfil$|$\hfil\cr\kern-3.01385pt\cr$\smile$\cr\kern-1.29167pt\cr}}
}}\displaylimits}^{a}_{E}B_{0}B_{1}$.

###### Proof.

If we consider $E,A,B_{0}$ and $B_{1}$ as in the statement we have to show
that $Diag_{B_{0}}(\langle A_{0}B_{0}\rangle)\cup Diag_{B_{1}}(\langle
A_{1}B_{1}\rangle)\cup\\{t(\overline{a})\centernot\in\langle
B_{0}B_{1}\rangle$ : for all terms $t$ such that $t(\overline{a})\centernot\in
E\\}$ is consistent, by compactness is it enough to consider finitely
generated structures $E,A,B_{0}$, and in that case we get the consistency by
just applying $6$. ∎

######  Definition 8.

We define an independence relation on small subsets of the monster model : for
$E\subseteq A,B$ say that $A\mathrel{\mathop{\vbox{
\hbox{\oalign{\kern-1.29167pt\cr\hfil$|$\hfil\cr\kern-3.01385pt\cr$\smile$\cr\kern-1.29167pt\cr}}
}}\displaylimits}^{\Gamma}_{E}B$ if the structure generated by $A$ and $B$,
written $\langle AB\rangle$ and the inclusions $i_{A}:A\rightarrow\langle
AB\rangle$, $i_{B}:B\rightarrow\langle AB\rangle$ satisfies the universal
property written as $A\oplus_{E}B$, i.e. every common extension of $A$ and $B$
factorises uniquely thought $\langle AB\rangle$.

######  Lemma 9.

Full existence : For every $E\subseteq A,B\subseteq\mathbb{M}$ such that
$A\cap B=E$ there is $A^{\prime}\equiv_{E}A$ such that
$A^{\prime}\mathrel{\mathop{\vbox{
\hbox{\oalign{\kern-1.29167pt\cr\hfil$|$\hfil\cr\kern-3.01385pt\cr$\smile$\cr\kern-1.29167pt\cr}}
}}\displaylimits}^{\Gamma}_{E}B$.

###### Proof.

We want to construct an $L$ structure $D$ extending $B$ and an embedding
$i_{A}:A\rightarrow D$ such that if $A^{\prime}:=i_{A}(A)$, $D=\langle
A^{\prime}B\rangle$ and $A^{\prime}\mathrel{\mathop{\vbox{
\hbox{\oalign{\kern-1.29167pt\cr\hfil$|$\hfil\cr\kern-3.01385pt\cr$\smile$\cr\kern-1.29167pt\cr}}
}}\displaylimits}^{\Gamma}_{E}B$. Consider the set of the $L$ terms with
parameters in $A,B,E$ viewed as constant symbols, to each of those terms $t$
we associate a variable $x_{t}$. We consider the equivalence relation $E$ of
$X$ defined by
$E[t(\overline{a},\overline{b},\overline{e}),t^{\prime}(\overline{a}^{\prime},\overline{b}^{\prime},\overline{e}^{\prime})]$
if and only if for some countable substructures $E^{\prime},A^{\prime}$ and
$B^{\prime}$ of $E$, $A$ and $B$ respectively these two terms computed in the
structure $A^{\prime}\oplus_{E^{\prime}}B^{\prime}$ are equal. To show the $E$
is transitive we need to look at the canonical map
$A^{\prime\prime}\oplus_{E^{\prime\prime}}B^{\prime\prime}$ to
$A^{\prime}\oplus_{E^{\prime}}B^{\prime}$ where $E^{\prime\prime}\subseteq
E^{\prime}$, $A^{\prime\prime}\subseteq A^{\prime}$,
$B^{\prime\prime}\subseteq B^{\prime}$ and $E^{\prime}=A^{\prime}\cap
B^{\prime}$. We describe it in the following diagram. This map assures us that
the system of the $A^{\prime}\oplus_{E^{\prime}}B^{\prime}$ we are considering
is directed and the structure we are constructing can also be presented as its
limit.

${{A^{\prime}\oplus_{E^{\prime}}B^{\prime}}}$${{A^{\prime}}}$${{A^{\prime\prime}\oplus_{E^{\prime\prime}}B^{\prime\prime}}}$${{B^{\prime}}}$${{A^{\prime\prime}}}$${{E^{\prime}}}$${{B^{\prime\prime}}}$${{E^{\prime\prime}}}$$\scriptstyle{\exists!\varphi}$

Now we give to $X/E$ the natural structure : for a function symbol $f$ set
$f^{D}([x_{\overline{t}}]):=[x_{f(\overline{t})}]$ and for a relation symbol
$R$ set $R^{D}([x_{\overline{t}}])$ if for some countable substructures
$E^{\prime},A^{\prime}$ and $B^{\prime}$ of $A$, $A$ and $B$ respectively the
terms $\overline{t}$ computed inside of
$A^{\prime}\oplus_{E^{\prime}}B^{\prime}$ satisfies $R$. This is well defined
by the same argument as previously and we get our structure $D$. By assumption
the diagram of $D$ over $B$ is finitely satisfied in $\mathbb{M}$, so we can
assume that $B\subseteq D\subseteq\mathbb{M}$, and the function that to $a\in
A$ associates the $E$ class of $x_{a}$ is an embedding from $A$ to $D$ over
$E$. So we have two embeddings $i_{A},i_{B}$ from $A$ and $B$ respectively to
$D$, with $i_{B}$ the inclusion, let $A^{\prime}=i_{A}(A)$ (so it is the set
of the classes of the terms in $A$). $D$ is generated by $A^{\prime}$ and $B$,
now if we have $D^{\prime}$, $\varphi_{A}$ and $\varphi_{B}$ as in the
statement the map $\varphi:D\rightarrow D^{\prime}$ which associates to
$x_{t(\overline{a},\overline{b},\overline{e})}$ the element
$t(\varphi_{A}(\overline{a}),\varphi_{B}(\overline{b}),\varphi_{E}(\overline{e}))$
is well defined and is a morphism (by the same argument as previously, we go
through $A^{\prime}\oplus_{E^{\prime}}B^{\prime}$), and it is naturally the
unique map satisfying the universal statement.∎

The difference with the previous free amalgam notion is that it was defined
for the elements of $K$ only, which corresponds to the countable substructures
of $\mathbb{M}$, whereas this notion is defined for small substructures. Also
since our class $K$ is axiomatisable every countable substructure of
$\mathbb{M}$ is inside of $K$ so there is no problem in considering the
structures $A^{\prime}\oplus_{E^{\prime}}B^{\prime}$ for any countable
substructures $E^{\prime},A^{\prime},B^{\prime}$ of $\mathbb{M}$.

######  Proposition 10.

The relation $A\mathrel{\mathop{\vbox{
\hbox{\oalign{\kern-1.29167pt\cr\hfil$|$\hfil\cr\kern-3.01385pt\cr$\smile$\cr\kern-1.29167pt\cr}}
}}\displaylimits}^{\Gamma}_{E}B$ defined for structures $E\subseteq A,B$ is
invariant, symmetric, satisfies existence, extension, monotonicity, finite
character, stationarity, base monotonicity, transitivity, implies
$\mathrel{\mathop{\vbox{
\hbox{\oalign{\kern-1.29167pt\cr\hfil$|$\hfil\cr\kern-3.01385pt\cr$\smile$\cr\kern-1.29167pt\cr}}
}}\displaylimits}^{a}$ and implies non dividing (and non-forking by
extension).

###### Proof.

By existence here we mean that for every extension $E\subseteq A$ we have
$A\mathrel{\mathop{\vbox{
\hbox{\oalign{\kern-1.29167pt\cr\hfil$|$\hfil\cr\kern-3.01385pt\cr$\smile$\cr\kern-1.29167pt\cr}}
}}\displaylimits}^{\Gamma}_{E}E$. The properties of symmetry, existence and
the fact that it implies $\mathrel{\mathop{\vbox{
\hbox{\oalign{\kern-1.29167pt\cr\hfil$|$\hfil\cr\kern-3.01385pt\cr$\smile$\cr\kern-1.29167pt\cr}}
}}\displaylimits}^{a}$ are immediate. For monotonicity assume that
$A\mathrel{\mathop{\vbox{
\hbox{\oalign{\kern-1.29167pt\cr\hfil$|$\hfil\cr\kern-3.01385pt\cr$\smile$\cr\kern-1.29167pt\cr}}
}}\displaylimits}^{\Gamma}_{E}B$ and consider $E\subseteq C\subseteq B$ a
substructure (this is enough by symmetry). We consider the free amalgam $D$ of
$A$ and $C$ and the associated embeddings $i_{A},i_{C}$. Now let us consider
the free amalgam of $D$ and $B$ over $C$ and its associated embeddings
$j_{D},j_{B}$. There is a unique morphism $\varphi_{D}:D\rightarrow\langle
AB\rangle$ such that $\varphi_{D}\cdot i_{A}=id_{A}$ and $\varphi_{D}\cdot
i_{C}=id_{C}$. There is a unique morphism
$\varphi_{D^{\prime}}:D^{\prime}\rightarrow\langle AB\rangle$ such that
$\varphi_{D^{\prime}}\cdot j_{D}=\varphi_{D}$ and $\varphi_{D^{\prime}}\cdot
j_{B}=id_{B}$. So $\varphi_{D^{\prime}}\cdot j_{D}\cdot i_{A}=id_{A}$, and we
get that $D^{\prime}$ is generated by the images of $A$ and $B$ and that
$\varphi_{D^{\prime}}$ is an isomorphism. We know that $D$ is generated by the
images of $A$ and $C$, and that it is sent to $\langle
AC\rangle\subseteq\langle AB\rangle$ by $j_{D}$, so $\varphi_{D^{\prime}}\cdot
j_{D}:D\rightarrow\langle AC\rangle$ is an isomorphism, and
$A\mathrel{\mathop{\vbox{
\hbox{\oalign{\kern-1.29167pt\cr\hfil$|$\hfil\cr\kern-3.01385pt\cr$\smile$\cr\kern-1.29167pt\cr}}
}}\displaylimits}^{\Gamma}_{E}C$.

${\langle
AB\rangle}$${D^{\prime}}$${D}$${B}$${A}$${C}$${E}$$\scriptstyle{\varphi_{D}}$$\scriptstyle{j_{D}}$$\scriptstyle{\varphi_{D^{\prime}}}$$\scriptstyle{j_{B}}$$\scriptstyle{i_{A}}$$\scriptstyle{i_{C}}$

For transitivity, given structures $E\subseteq A,C$, $C\subseteq B$ such that
$A\mathrel{\mathop{\vbox{
\hbox{\oalign{\kern-1.29167pt\cr\hfil$|$\hfil\cr\kern-3.01385pt\cr$\smile$\cr\kern-1.29167pt\cr}}
}}\displaylimits}^{\Gamma}_{E}C$ and $\langle AC\rangle\mathrel{\mathop{\vbox{
\hbox{\oalign{\kern-1.29167pt\cr\hfil$|$\hfil\cr\kern-3.01385pt\cr$\smile$\cr\kern-1.29167pt\cr}}
}}\displaylimits}^{\Gamma}_{C}B$ we want to show that
$A\mathrel{\mathop{\vbox{
\hbox{\oalign{\kern-1.29167pt\cr\hfil$|$\hfil\cr\kern-3.01385pt\cr$\smile$\cr\kern-1.29167pt\cr}}
}}\displaylimits}^{\Gamma}_{E}B$. For this we use the universal property
Proposition 10. Consider a structure $D$ and two embeddings
$\varphi_{A},\varphi_{B}$ that coincide in $E$ from $A$ and $B$ respectively
to $D$. Then the embeddings $\varphi_{A}$ and
$\mathchoice{{\varphi_{B}\,\smash{\vrule
height=3.44444pt,depth=1.65279pt}}_{\,C}}{{\varphi_{B}\,\smash{\vrule
height=3.44444pt,depth=1.65279pt}}_{\,C}}{{\varphi_{B}\,\smash{\vrule
height=2.41112pt,depth=1.16167pt}}_{\,C}}{{\varphi_{B}\,\smash{\vrule
height=1.72221pt,depth=1.16167pt}}_{\,C}}$ coincide over $E$, and by
assumption there is a unique map $\varphi_{\langle AC\rangle}:\langle
AC\rangle\rightarrow D$ such that $\mathchoice{{\varphi_{B}\,\smash{\vrule
height=3.44444pt,depth=1.65279pt}}_{\,C}}{{\varphi_{B}\,\smash{\vrule
height=3.44444pt,depth=1.65279pt}}_{\,C}}{{\varphi_{B}\,\smash{\vrule
height=2.41112pt,depth=1.16167pt}}_{\,C}}{{\varphi_{B}\,\smash{\vrule
height=1.72221pt,depth=1.16167pt}}_{\,C}}=\mathchoice{{\varphi_{\langle
AC\rangle}\,\smash{\vrule
height=3.44444pt,depth=4.25003pt}}_{\,C}}{{\varphi_{\langle
AC\rangle}\,\smash{\vrule
height=3.44444pt,depth=4.25003pt}}_{\,C}}{{\varphi_{\langle
AC\rangle}\,\smash{\vrule
height=2.41112pt,depth=4.25003pt}}_{\,C}}{{\varphi_{\langle
AC\rangle}\,\smash{\vrule height=1.72221pt,depth=4.25003pt}}_{\,C}}$ and
$\varphi_{A}=\mathchoice{{\varphi_{\langle AC\rangle}\,\smash{\vrule
height=3.44444pt,depth=4.25003pt}}_{\,A}}{{\varphi_{\langle
AC\rangle}\,\smash{\vrule
height=3.44444pt,depth=4.25003pt}}_{\,A}}{{\varphi_{\langle
AC\rangle}\,\smash{\vrule
height=2.41112pt,depth=4.25003pt}}_{\,A}}{{\varphi_{\langle
AC\rangle}\,\smash{\vrule height=1.72221pt,depth=4.25003pt}}_{\,A}}$. Now we
consider the two maps $\varphi_{\langle AC\rangle}$ and $\varphi_{B}$ which
coincide over $C$. Since $\langle AC\rangle\mathrel{\mathop{\vbox{
\hbox{\oalign{\kern-1.29167pt\cr\hfil$|$\hfil\cr\kern-3.01385pt\cr$\smile$\cr\kern-1.29167pt\cr}}
}}\displaylimits}_{C}^{\Gamma}B$ there is a unique map $\varphi:\langle\langle
AC\rangle B\rangle=\langle AB\rangle\rightarrow D$ such that
$\mathchoice{{\varphi\,\smash{\vrule
height=3.44444pt,depth=1.65279pt}}_{\,B}}{{\varphi\,\smash{\vrule
height=3.44444pt,depth=1.65279pt}}_{\,B}}{{\varphi\,\smash{\vrule
height=2.41112pt,depth=1.16167pt}}_{\,B}}{{\varphi\,\smash{\vrule
height=1.72221pt,depth=1.16167pt}}_{\,B}}=\varphi_{B}$ and
$\mathchoice{{\varphi\,\smash{\vrule
height=3.44444pt,depth=4.25003pt}}_{\,\langle
AC\rangle}}{{\varphi\,\smash{\vrule
height=3.44444pt,depth=4.25003pt}}_{\,\langle
AC\rangle}}{{\varphi\,\smash{\vrule
height=2.41112pt,depth=4.25003pt}}_{\,\langle
AC\rangle}}{{\varphi\,\smash{\vrule
height=1.72221pt,depth=4.25003pt}}_{\,\langle AC\rangle}}=\varphi_{\langle
AC\rangle}$, so in particular $\mathchoice{{\varphi\,\smash{\vrule
height=3.44444pt,depth=1.65279pt}}_{\,A}}{{\varphi\,\smash{\vrule
height=3.44444pt,depth=1.65279pt}}_{\,A}}{{\varphi\,\smash{\vrule
height=2.41112pt,depth=1.16167pt}}_{\,A}}{{\varphi\,\smash{\vrule
height=1.72221pt,depth=1.16167pt}}_{\,A}}=\varphi_{A}$. This proof consist
just in saying that the square formed by two adjacent pushouts forms a
pushout.

${D}$${\langle AB\rangle}$${\langle
AC\rangle}$${B}$${A}$${C}$${E}$$\scriptstyle{\exists!\varphi}$$\scriptstyle{\exists!\varphi_{\langle
AC\rangle}}$$\scriptstyle{\varphi_{B}}$$\scriptstyle{\varphi_{A}}$

For extension and stationarity, given structures $E\subseteq A,C$, $C\subseteq
B$ such that $A\mathrel{\mathop{\vbox{
\hbox{\oalign{\kern-1.29167pt\cr\hfil$|$\hfil\cr\kern-3.01385pt\cr$\smile$\cr\kern-1.29167pt\cr}}
}}\displaylimits}^{\Gamma}_{E}C$ we can form the free amalgam of $\langle
AB\rangle$ and $C$ over $B$ inside of $\mathbb{M}$ thanks to Lemma 9, this
allows us to find some $A^{\prime}$ such that $\langle
A^{\prime}C\rangle\cong\langle AC\rangle$ (since the free amalgam defines a
unique isomorphism type), so by Q.E. $A^{\prime}\equiv_{B}A$ and $\langle
A^{\prime}B\rangle\mathrel{\mathop{\vbox{
\hbox{\oalign{\kern-1.29167pt\cr\hfil$|$\hfil\cr\kern-3.01385pt\cr$\smile$\cr\kern-1.29167pt\cr}}
}}\displaylimits}^{\Gamma}_{B}C$, so $A^{\prime}\mathrel{\mathop{\vbox{
\hbox{\oalign{\kern-1.29167pt\cr\hfil$|$\hfil\cr\kern-3.01385pt\cr$\smile$\cr\kern-1.29167pt\cr}}
}}\displaylimits}^{\Gamma}_{E}C$ by transitivity, and this gives us a unique
type over $C$ for the same reason. Finite character is also a direct
consequence of the universal property : assume that
$A\centernot\mathrel{\mathop{\vbox{
\hbox{\oalign{\kern-1.29167pt\cr\hfil$|$\hfil\cr\kern-3.01385pt\cr$\smile$\cr\kern-1.29167pt\cr}}
}}\displaylimits}^{K}_{E}B$, so there is a structure $D$ and two embeddings
$\varphi_{A}$, $\varphi_{B}$ from $A$ and $B$ respectively to $D$ that
coincide in $E$ such that there is no map $\varphi:\langle
AB\rangle\rightarrow D$ satisfying $\varphi_{A}=\varphi|_{A}$ and
$\varphi_{B}=\varphi|_{B}$. This is equivalent to saying that the function
that to an element $t(\overline{a},\overline{b})\in\langle AB\rangle$
associates $t(\varphi_{A}(\overline{a}),\varphi_{B}(\overline{b}))$ is not
well defined (since this function is the only candidate) or that it is not a
morphism. In either of those cases we can find some finite tuples
$\overline{a}\in A$, $\overline{b}\in B$ such that the same thing is happening
with the function that to an element
$t(\overline{a}^{\prime},\overline{b}^{\prime})\in\langle
E\overline{a}\overline{b}\rangle$ associates
$t(\varphi_{A}(\overline{a}^{\prime}),\varphi_{B}(\overline{b}^{\prime}))$, so
$\overline{a}\centernot\mathrel{\mathop{\vbox{
\hbox{\oalign{\kern-1.29167pt\cr\hfil$|$\hfil\cr\kern-3.01385pt\cr$\smile$\cr\kern-1.29167pt\cr}}
}}\displaylimits}^{K}_{E}\overline{b}$.

For base monotonicity, given structures $E\subseteq A,C$, $C\subseteq B$ such
that $A\mathrel{\mathop{\vbox{
\hbox{\oalign{\kern-1.29167pt\cr\hfil$|$\hfil\cr\kern-3.01385pt\cr$\smile$\cr\kern-1.29167pt\cr}}
}}\displaylimits}^{\Gamma}_{E}B$ we want to show that $\langle
AC\rangle\mathrel{\mathop{\vbox{
\hbox{\oalign{\kern-1.29167pt\cr\hfil$|$\hfil\cr\kern-3.01385pt\cr$\smile$\cr\kern-1.29167pt\cr}}
}}\displaylimits}^{\Gamma}_{C}B$. We take the same notations as for
transitivity. As previously $A\mathrel{\mathop{\vbox{
\hbox{\oalign{\kern-1.29167pt\cr\hfil$|$\hfil\cr\kern-3.01385pt\cr$\smile$\cr\kern-1.29167pt\cr}}
}}\displaylimits}^{\Gamma}_{E}C$ (by monotonicity this time). For this we use
the universal property Proposition 10. Consider a structure $D$ and two
embeddings $\varphi_{\langle AC\rangle},\varphi_{B}$ from $\langle AC\rangle$
and $B$ respectively to $D$ that coincide in $C$. Then the embeddings
$\mathchoice{{\varphi_{\langle AC\rangle}\,\smash{\vrule
height=3.44444pt,depth=4.25003pt}}_{\,A}}{{\varphi_{\langle
AC\rangle}\,\smash{\vrule
height=3.44444pt,depth=4.25003pt}}_{\,A}}{{\varphi_{\langle
AC\rangle}\,\smash{\vrule
height=2.41112pt,depth=4.25003pt}}_{\,A}}{{\varphi_{\langle
AC\rangle}\,\smash{\vrule height=1.72221pt,depth=4.25003pt}}_{\,A}}$ and
$\varphi_{B}$ coincide over $E$, by assumption there is a unique map
$\varphi:\langle AB\rangle=\langle\langle AC\rangle B\rangle\rightarrow D$
such that $\mathchoice{{\varphi\,\smash{\vrule
height=3.44444pt,depth=1.65279pt}}_{\,A}}{{\varphi\,\smash{\vrule
height=3.44444pt,depth=1.65279pt}}_{\,A}}{{\varphi\,\smash{\vrule
height=2.41112pt,depth=1.16167pt}}_{\,A}}{{\varphi\,\smash{\vrule
height=1.72221pt,depth=1.16167pt}}_{\,A}}=\mathchoice{{\varphi_{\langle
AC\rangle}\,\smash{\vrule
height=3.44444pt,depth=4.25003pt}}_{\,A}}{{\varphi_{\langle
AC\rangle}\,\smash{\vrule
height=3.44444pt,depth=4.25003pt}}_{\,A}}{{\varphi_{\langle
AC\rangle}\,\smash{\vrule
height=2.41112pt,depth=4.25003pt}}_{\,A}}{{\varphi_{\langle
AC\rangle}\,\smash{\vrule height=1.72221pt,depth=4.25003pt}}_{\,A}}$ and
$\mathchoice{{\varphi\,\smash{\vrule
height=3.44444pt,depth=1.65279pt}}_{\,B}}{{\varphi\,\smash{\vrule
height=3.44444pt,depth=1.65279pt}}_{\,B}}{{\varphi\,\smash{\vrule
height=2.41112pt,depth=1.16167pt}}_{\,B}}{{\varphi\,\smash{\vrule
height=1.72221pt,depth=1.16167pt}}_{\,B}}=\varphi_{B}$. Then we also have that
$\mathchoice{{\varphi\,\smash{\vrule
height=3.44444pt,depth=1.65279pt}}_{\,C}}{{\varphi\,\smash{\vrule
height=3.44444pt,depth=1.65279pt}}_{\,C}}{{\varphi\,\smash{\vrule
height=2.41112pt,depth=1.16167pt}}_{\,C}}{{\varphi\,\smash{\vrule
height=1.72221pt,depth=1.16167pt}}_{\,C}}=\mathchoice{{\varphi_{B}\,\smash{\vrule
height=3.44444pt,depth=1.65279pt}}_{\,C}}{{\varphi_{B}\,\smash{\vrule
height=3.44444pt,depth=1.65279pt}}_{\,C}}{{\varphi_{B}\,\smash{\vrule
height=2.41112pt,depth=1.16167pt}}_{\,C}}{{\varphi_{B}\,\smash{\vrule
height=1.72221pt,depth=1.16167pt}}_{\,C}}=\mathchoice{{\varphi_{\langle
AC\rangle}\,\smash{\vrule
height=3.44444pt,depth=4.25003pt}}_{\,C}}{{\varphi_{\langle
AC\rangle}\,\smash{\vrule
height=3.44444pt,depth=4.25003pt}}_{\,C}}{{\varphi_{\langle
AC\rangle}\,\smash{\vrule
height=2.41112pt,depth=4.25003pt}}_{\,C}}{{\varphi_{\langle
AC\rangle}\,\smash{\vrule height=1.72221pt,depth=4.25003pt}}_{\,C}}$, so since
$A\mathrel{\mathop{\vbox{
\hbox{\oalign{\kern-1.29167pt\cr\hfil$|$\hfil\cr\kern-3.01385pt\cr$\smile$\cr\kern-1.29167pt\cr}}
}}\displaylimits}^{\Gamma}_{E}C$ we have that
$\mathchoice{{\varphi\,\smash{\vrule
height=3.44444pt,depth=4.25003pt}}_{\,\langle
AC\rangle}}{{\varphi\,\smash{\vrule
height=3.44444pt,depth=4.25003pt}}_{\,\langle
AC\rangle}}{{\varphi\,\smash{\vrule
height=2.41112pt,depth=4.25003pt}}_{\,\langle
AC\rangle}}{{\varphi\,\smash{\vrule
height=1.72221pt,depth=4.25003pt}}_{\,\langle AC\rangle}}=\varphi_{\langle
AC\rangle}$. This proof consist just in saying that if the square formed by
two adjacent squares is a pushout and the left one also is then the right one
is also a pushout.

${D}$${\langle AB\rangle}$${\langle
AC\rangle}$${B}$${A}$${C}$${E}$$\scriptstyle{\exists!\varphi}$$\scriptstyle{\varphi_{\langle
AC\rangle}}$$\scriptstyle{\varphi_{B}}$$\scriptstyle{\varphi_{A}}$

Now we want to show that $A\mathrel{\mathop{\vbox{
\hbox{\oalign{\kern-1.29167pt\cr\hfil$|$\hfil\cr\kern-3.01385pt\cr$\smile$\cr\kern-1.29167pt\cr}}
}}\displaylimits}^{\Gamma}_{E}B$ implies that $A\mathrel{\mathop{\vbox{
\hbox{\oalign{\kern-1.29167pt\cr\hfil$|$\hfil\cr\kern-3.01385pt\cr$\smile$\cr\kern-1.29167pt\cr}}
}}\displaylimits}^{d}_{E}B$. Let $(B_{i})_{i<\omega}$ be an $E$-indiscernible
sequence. We begin by showing that we can assume that
$B_{i}\mathrel{\mathop{\vbox{
\hbox{\oalign{\kern-1.29167pt\cr\hfil$|$\hfil\cr\kern-3.01385pt\cr$\smile$\cr\kern-1.29167pt\cr}}
}}\displaylimits}^{a}_{E}B_{<i}$ for every $i<\omega$, then the proof is
straightforward using Proposition 12. Begin by adding a negative part to the
sequence $(B_{k})_{k\in\omega+\omega}$, by extension we can assume that
$A\mathrel{\mathop{\vbox{
\hbox{\oalign{\kern-1.29167pt\cr\hfil$|$\hfil\cr\kern-3.01385pt\cr$\smile$\cr\kern-1.29167pt\cr}}
}}\displaylimits}^{\Gamma}_{E}\langle(B_{k})_{k\in\omega+\omega}\rangle$. Now
set $E^{\prime}=\langle(B_{k})_{k<0}\rangle$ the structure generated by the
negative part. Consider the structures $B^{\prime}_{i}=\langle
E^{\prime}B_{i}\rangle$ for $i<\omega$. They form an
$E^{\prime}$-indiscernible sequence, define $A^{\prime}=\langle
E^{\prime}A\rangle$, by base monotonicity we have
$A^{\prime}\mathrel{\mathop{\vbox{
\hbox{\oalign{\kern-1.29167pt\cr\hfil$|$\hfil\cr\kern-3.01385pt\cr$\smile$\cr\kern-1.29167pt\cr}}
}}\displaylimits}^{\Gamma}_{E^{\prime}}\langle(B^{\prime}_{i})_{i<\omega}\rangle$.
Now if $b_{i}\in B^{\prime}_{i}$ is also in $\langle B^{\prime}_{<i}\rangle$
then it has to be in $E^{\prime}$ by indiscernibility of
$(B_{k})_{k\in\omega+\omega}$ (this is the point of adding the negative part :
algebraic relations happening in the sequence are already happening inside of
the negative part). So, replacing $A$ by $A^{\prime}$, $E$ by $E^{\prime}$ and
$(B_{i})_{i<\omega}$ by $(B^{\prime}_{i})_{i<\omega}$ we get our first point.

Now assume that $B_{i}\mathrel{\mathop{\vbox{
\hbox{\oalign{\kern-1.29167pt\cr\hfil$|$\hfil\cr\kern-3.01385pt\cr$\smile$\cr\kern-1.29167pt\cr}}
}}\displaylimits}^{a}_{E}B_{<i}$ for every $i<\omega$. Since
$B_{0}\equiv_{E}B_{i}$ these is some $A_{i}$ such that
$AB_{0}\equiv_{E}A_{i}B_{i}$. We define by induction on $i<\omega$ a sequence
of structures $A^{\prime}_{i}$ such that
$A^{\prime}_{i}\mathrel{\mathop{\vbox{
\hbox{\oalign{\kern-1.29167pt\cr\hfil$|$\hfil\cr\kern-3.01385pt\cr$\smile$\cr\kern-1.29167pt\cr}}
}}\displaylimits}^{a}_{E}B_{\leq i}$ and for all $j\leq i$
$AB_{0}\equiv_{E}A^{\prime}_{i}B_{j}$. Let $A_{0}=A^{\prime}_{0}=A$. If
$A^{\prime}_{i}$ has been defined we have
$A^{\prime}_{i}\mathrel{\mathop{\vbox{
\hbox{\oalign{\kern-1.29167pt\cr\hfil$|$\hfil\cr\kern-3.01385pt\cr$\smile$\cr\kern-1.29167pt\cr}}
}}\displaylimits}^{a}_{E}B_{\leq i}$, $A_{i+1}\mathrel{\mathop{\vbox{
\hbox{\oalign{\kern-1.29167pt\cr\hfil$|$\hfil\cr\kern-3.01385pt\cr$\smile$\cr\kern-1.29167pt\cr}}
}}\displaylimits}^{a}_{E}B_{i+1}$, $B_{i+1}\mathrel{\mathop{\vbox{
\hbox{\oalign{\kern-1.29167pt\cr\hfil$|$\hfil\cr\kern-3.01385pt\cr$\smile$\cr\kern-1.29167pt\cr}}
}}\displaylimits}^{a}_{E}B_{\leq i}$ and $A^{\prime}_{i}\equiv_{E}A_{i}$, so
by Proposition 12 we can find a suiting $A^{\prime}_{i+1}$. By compactness
there is an $A^{\prime}$ such that $A^{\prime}B_{i}\equiv_{E}AB_{0}$, so
$A\mathrel{\mathop{\vbox{
\hbox{\oalign{\kern-1.29167pt\cr\hfil$|$\hfil\cr\kern-3.01385pt\cr$\smile$\cr\kern-1.29167pt\cr}}
}}\displaylimits}^{d}_{E}B$.∎

######  Corollary 11.

$T$ has existence, is NSOP1 and $\mathrel{\mathop{\vbox{
\hbox{\oalign{\kern-1.29167pt\cr\hfil$|$\hfil\cr\kern-3.01385pt\cr$\smile$\cr\kern-1.29167pt\cr}}
}}\displaylimits}^{K}=\mathrel{\mathop{\vbox{
\hbox{\oalign{\kern-1.29167pt\cr\hfil$|$\hfil\cr\kern-3.01385pt\cr$\smile$\cr\kern-1.29167pt\cr}}
}}\displaylimits}^{a}$ over arbitrary sets.

###### Proof.

Existence is a direct consequence of Proposition 10. We use the Kim-Pillay
theorem for Kim-forking [9, Theorem 5.1], by [18, Lemma 2.7]
$\mathrel{\mathop{\vbox{
\hbox{\oalign{\kern-1.29167pt\cr\hfil$|$\hfil\cr\kern-3.01385pt\cr$\smile$\cr\kern-1.29167pt\cr}}
}}\displaylimits}^{a}$ satisfies extension, existence, monotonicity, symmetry,
strong finite character, and witnessing. By Lemma 7 it satisfies the
amalgamation theorem, and it is clear that it satisfies transitivity. We now
show that it satisfies the local character condition. Let $\overline{a}$ be a
finite tuple, $\kappa$ an uncountable regular cardinal, $(A_{i})_{i<\kappa}$ a
continuous increasing sequence of structure of cardinal smaller than $\kappa$
and $A_{\kappa}=\bigcup\limits_{i<\kappa}A_{i}$. We want to show that there is
an $i<\kappa$ such that $\langle
A_{i}\overline{a}\rangle\mathrel{\mathop{\vbox{
\hbox{\oalign{\kern-1.29167pt\cr\hfil$|$\hfil\cr\kern-3.01385pt\cr$\smile$\cr\kern-1.29167pt\cr}}
}}\displaylimits}^{a}_{A_{i}}A_{\kappa}$. We choose some $i_{0}<\kappa$. If
$\langle A_{i_{0}}\overline{a}\rangle\centernot\mathrel{\mathop{\vbox{
\hbox{\oalign{\kern-1.29167pt\cr\hfil$|$\hfil\cr\kern-3.01385pt\cr$\smile$\cr\kern-1.29167pt\cr}}
}}\displaylimits}^{a}_{A_{i_{0}}}A_{\kappa}$ the set $\langle
A_{i_{0}}\overline{a}\rangle\cap A_{\kappa}$ is non empty and has cardinal
smaller than $\kappa$, so it is contained in $A_{i_{1}}$ for some
$i_{1}>i_{0}$. We iterate this process to define a sequence
$(i_{j})_{j<\omega}$ and set $i=\bigcup\limits_{j<\omega}i_{j}$ which is
smaller than $\kappa$ by regularity. Now if $a\in\langle
A_{i}\overline{a}\rangle\cup A_{\kappa}$ it is in $\langle
A_{i_{j}}\overline{a}\rangle\cup A_{\kappa}$ for some $j<\omega$ so by
construction it is inside of $A_{i_{j+1}}\subseteq A_{i}$. ∎

######  Remark 12.

By [17, Theorem 2.7] we know that when applying the amalgamation theorem to
$E,A_{0},A_{1},B_{0},B_{1}$ (as in the statement of the amalgamation theorem)
for $\mathrel{\mathop{\vbox{
\hbox{\oalign{\kern-1.29167pt\cr\hfil$|$\hfil\cr\kern-3.01385pt\cr$\smile$\cr\kern-1.29167pt\cr}}
}}\displaylimits}^{K}$ in an NSOP1 theory with existence we can take $A$ such
that $AB_{i}\mathrel{\mathop{\vbox{
\hbox{\oalign{\kern-1.29167pt\cr\hfil$|$\hfil\cr\kern-3.01385pt\cr$\smile$\cr\kern-1.29167pt\cr}}
}}\displaylimits}_{B_{i}}B_{0}B_{1}$ for $i=0,1$. In our setting by Lemma 7
and ($3$ Amalgamation for algebraically independent extension) we also have
$AB_{0}\mathrel{\mathop{\vbox{
\hbox{\oalign{\kern-1.29167pt\cr\hfil$|$\hfil\cr\kern-3.01385pt\cr$\smile$\cr\kern-1.29167pt\cr}}
}}\displaylimits}^{K}_{A}AB_{1}$, however i do not know if this is true in
general. Also notice that in order to get this result we only need $1-5$ and
Lemma 7 to hold and not the full generality of ($3$ Amalgamation for
algebraically independent extension).

The following proposition is a general fact for theories such that algebraic
independence satisfies the independence theorem, a proof can be found in [18,
Proposition 3.22]. This fact is part of some more general results about
deducting weak elimination of imaginaries from the properties of some
independence relations (see [12, Section 4.4] for example).

######  Proposition 13.

The theory $T$ has weak elimination of imaginaries.

######  Definition 14.

We define $M$-independence $\mathrel{\mathop{\vbox{
\hbox{\oalign{\kern-1.29167pt\cr\hfil$|$\hfil\cr\kern-3.01385pt\cr$\smile$\cr\kern-1.29167pt\cr}}
}}\displaylimits}^{M}$ by “forcing base monotonicity” on
$\mathrel{\mathop{\vbox{
\hbox{\oalign{\kern-1.29167pt\cr\hfil$|$\hfil\cr\kern-3.01385pt\cr$\smile$\cr\kern-1.29167pt\cr}}
}}\displaylimits}^{a}$ : $A\mathrel{\mathop{\vbox{
\hbox{\oalign{\kern-1.29167pt\cr\hfil$|$\hfil\cr\kern-3.01385pt\cr$\smile$\cr\kern-1.29167pt\cr}}
}}\displaylimits}^{M}_{E}B$ if $A\mathrel{\mathop{\vbox{
\hbox{\oalign{\kern-1.29167pt\cr\hfil$|$\hfil\cr\kern-3.01385pt\cr$\smile$\cr\kern-1.29167pt\cr}}
}}\displaylimits}^{a}_{C}B$ for every $E\subseteq C\subseteq B$.

######  Proposition 15.

In $\mathbb{M}$ we have that $\mathrel{\mathop{\vbox{
\hbox{\oalign{\kern-1.29167pt\cr\hfil$|$\hfil\cr\kern-3.01385pt\cr$\smile$\cr\kern-1.29167pt\cr}}
}}\displaylimits}^{M}$ satisfies extension, so $\mathrel{\mathop{\vbox{
\hbox{\oalign{\kern-1.29167pt\cr\hfil$|$\hfil\cr\kern-3.01385pt\cr$\smile$\cr\kern-1.29167pt\cr}}
}}\displaylimits}^{f}=\mathrel{\mathop{\vbox{
\hbox{\oalign{\kern-1.29167pt\cr\hfil$|$\hfil\cr\kern-3.01385pt\cr$\smile$\cr\kern-1.29167pt\cr}}
}}\displaylimits}^{d}=\mathrel{\mathop{\vbox{
\hbox{\oalign{\kern-1.29167pt\cr\hfil$|$\hfil\cr\kern-3.01385pt\cr$\smile$\cr\kern-1.29167pt\cr}}
}}\displaylimits}^{M}$ over arbitrary sets.

###### Proof.

Consider $E\subseteq A,B\subseteq\mathbb{M}$ such that
$A\mathrel{\mathop{\vbox{
\hbox{\oalign{\kern-1.29167pt\cr\hfil$|$\hfil\cr\kern-3.01385pt\cr$\smile$\cr\kern-1.29167pt\cr}}
}}\displaylimits}^{M}_{E}B$ and $B\subseteq B^{\prime}$. We can find
$A^{\prime}$ such that $A^{\prime}\equiv_{B}A$ and $\langle
A^{\prime}B\rangle\mathrel{\mathop{\vbox{
\hbox{\oalign{\kern-1.29167pt\cr\hfil$|$\hfil\cr\kern-3.01385pt\cr$\smile$\cr\kern-1.29167pt\cr}}
}}\displaylimits}^{\Gamma}_{B}B^{\prime}$, let us write $D^{\prime}=\langle
A^{\prime}B^{\prime}\rangle$. We show that $A^{\prime}\mathrel{\mathop{\vbox{
\hbox{\oalign{\kern-1.29167pt\cr\hfil$|$\hfil\cr\kern-3.01385pt\cr$\smile$\cr\kern-1.29167pt\cr}}
}}\displaylimits}^{M}_{E}B^{\prime}$. Consider $C^{\prime}\subseteq
B^{\prime}$ and define $C=C^{\prime}\cap B$, we want to show that $\langle
A^{\prime}C^{\prime}\rangle\cap B^{\prime}=C^{\prime}$. By $6$, since $\langle
A^{\prime}C\rangle\cap C^{\prime}=C$, $\langle A^{\prime}C\rangle\cap B=C$
(because $A\mathrel{\mathop{\vbox{
\hbox{\oalign{\kern-1.29167pt\cr\hfil$|$\hfil\cr\kern-3.01385pt\cr$\smile$\cr\kern-1.29167pt\cr}}
}}\displaylimits}^{M}_{E}B$) and $B\cap C^{\prime}=C$, the following diagram
with the solid arrows, where $\langle A^{\prime}C^{\prime}\rangle$ is the
structure generated by $A^{\prime}$ and $C^{\prime}$ in $D^{\prime}$, can be
completed with the dashed arrows and a structure $D_{C^{\prime}}$ such that
$\varphi_{B^{\prime}}(B^{\prime})\cap\varphi_{\langle
A^{\prime}C^{\prime}\rangle}(\langle
A^{\prime}C^{\prime}\rangle)=\varphi_{B^{\prime}}(C^{\prime})=\varphi_{\langle
A^{\prime}C^{\prime}\rangle}(C^{\prime})$.

${\langle
A^{\prime}C^{\prime}\rangle}$${D^{\prime}}$${D_{C^{\prime}}}$${\langle
A^{\prime}C\rangle}$${\langle
A^{\prime}B\rangle}$${C^{\prime}}$${B^{\prime}}$${C}$${B}$$\scriptstyle{\subseteq}$$\scriptstyle{\varphi_{\langle
A^{\prime}C^{\prime}\rangle}}$$\scriptstyle{\exists!\varphi}$$\scriptstyle{\subseteq}$$\scriptstyle{\varphi_{\langle
A^{\prime}B\rangle}}$$\scriptstyle{\subseteq}$$\scriptstyle{\varphi_{B^{\prime}}}$

There is a unique map $\varphi$ such that $\varphi|_{\langle
A^{\prime}B\rangle}=\varphi_{\langle A^{\prime}B\rangle}$ and
$\varphi|_{B^{\prime}}=\varphi_{B^{\prime}}$. The two maps $\varphi_{\langle
A^{\prime}C^{\prime}\rangle}$ and $\varphi|_{\langle
A^{\prime}C^{\prime}\rangle}$ coincide on the sets $C^{\prime}$ and $\langle
A^{\prime}C\rangle$, and these sets generates $\langle
A^{\prime}C^{\prime}\rangle$, so $\varphi|_{\langle
A^{\prime}C^{\prime}\rangle}=\varphi_{\langle A^{\prime}C^{\prime}\rangle}$.
If we have some equality between two terms
$t(a^{\prime},c^{\prime})=t^{\prime}(b^{\prime})$ for $a^{\prime}\in
A^{\prime}$, $c^{\prime}\in C^{\prime}$ and $b^{\prime}\in B^{\prime}$ inside
of $D^{\prime}$ then by applying $\varphi$ we get that $\varphi_{\langle
A^{\prime}C^{\prime}\rangle}(t(a^{\prime},c^{\prime}))=\varphi_{B^{\prime}}(t^{\prime}(b^{\prime}))$,
so by $6$ there is a $c\in C^{\prime}$ such that $\varphi_{\langle
A^{\prime}C^{\prime}\rangle}(c)=\varphi_{\langle
A^{\prime}C^{\prime}\rangle}(t(a^{\prime},c^{\prime}))$ and
$\varphi_{B^{\prime}}(c)=\varphi_{B^{\prime}}(t^{\prime}(b^{\prime}))$. Since
the maps $\varphi_{B^{\prime}}$ and $\varphi_{\langle
A^{\prime}C^{\prime}\rangle}$ are embeddings we get that
$t(a^{\prime},c^{\prime})=t^{\prime}(b^{\prime})\in B^{\prime}$, so $\langle
A^{\prime}C^{\prime}\rangle\cup B^{\prime}=C^{\prime}$.

Now for the second part of the statement it is always true that
$\mathrel{\mathop{\vbox{
\hbox{\oalign{\kern-1.29167pt\cr\hfil$|$\hfil\cr\kern-3.01385pt\cr$\smile$\cr\kern-1.29167pt\cr}}
}}\displaylimits}^{d}\implies\mathrel{\mathop{\vbox{
\hbox{\oalign{\kern-1.29167pt\cr\hfil$|$\hfil\cr\kern-3.01385pt\cr$\smile$\cr\kern-1.29167pt\cr}}
}}\displaylimits}^{M}$. To show that $\mathrel{\mathop{\vbox{
\hbox{\oalign{\kern-1.29167pt\cr\hfil$|$\hfil\cr\kern-3.01385pt\cr$\smile$\cr\kern-1.29167pt\cr}}
}}\displaylimits}^{M}\implies\mathrel{\mathop{\vbox{
\hbox{\oalign{\kern-1.29167pt\cr\hfil$|$\hfil\cr\kern-3.01385pt\cr$\smile$\cr\kern-1.29167pt\cr}}
}}\displaylimits}^{f}$ it is sufficient to show that $\mathrel{\mathop{\vbox{
\hbox{\oalign{\kern-1.29167pt\cr\hfil$|$\hfil\cr\kern-3.01385pt\cr$\smile$\cr\kern-1.29167pt\cr}}
}}\displaylimits}^{M}\implies\mathrel{\mathop{\vbox{
\hbox{\oalign{\kern-1.29167pt\cr\hfil$|$\hfil\cr\kern-3.01385pt\cr$\smile$\cr\kern-1.29167pt\cr}}
}}\displaylimits}^{d}$ by extension, and for this we can run the same proof as
the one of $\mathrel{\mathop{\vbox{
\hbox{\oalign{\kern-1.29167pt\cr\hfil$|$\hfil\cr\kern-3.01385pt\cr$\smile$\cr\kern-1.29167pt\cr}}
}}\displaylimits}^{\Gamma}\implies\mathrel{\mathop{\vbox{
\hbox{\oalign{\kern-1.29167pt\cr\hfil$|$\hfil\cr\kern-3.01385pt\cr$\smile$\cr\kern-1.29167pt\cr}}
}}\displaylimits}^{d}$ in Proposition 10 : assume that
$A\mathrel{\mathop{\vbox{
\hbox{\oalign{\kern-1.29167pt\cr\hfil$|$\hfil\cr\kern-3.01385pt\cr$\smile$\cr\kern-1.29167pt\cr}}
}}\displaylimits}^{M}_{E}B_{0}$ and consider an indiscernible sequence
$(B_{i})_{i<\omega}$. Extend it to $(B_{i})_{i<\omega+\omega}$. By extension
we can assume that $A\mathrel{\mathop{\vbox{
\hbox{\oalign{\kern-1.29167pt\cr\hfil$|$\hfil\cr\kern-3.01385pt\cr$\smile$\cr\kern-1.29167pt\cr}}
}}\displaylimits}^{M}_{E}(B_{i})_{i<\omega+\omega}$ and set
$E^{\prime}=\langle B_{<0}$, $A^{\prime}=\langle AE^{\prime}\rangle$, and
$B^{\prime}_{i}=\langle E^{\prime}B_{i}$ for all $i<\omega$. Then
$A^{\prime}\mathrel{\mathop{\vbox{
\hbox{\oalign{\kern-1.29167pt\cr\hfil$|$\hfil\cr\kern-3.01385pt\cr$\smile$\cr\kern-1.29167pt\cr}}
}}\displaylimits}^{M}_{E^{\prime}}B^{\prime}_{0}$,
$B^{\prime}_{i}\mathrel{\mathop{\vbox{
\hbox{\oalign{\kern-1.29167pt\cr\hfil$|$\hfil\cr\kern-3.01385pt\cr$\smile$\cr\kern-1.29167pt\cr}}
}}\displaylimits}^{a}_{E^{\prime}}B^{\prime}_{<i}$ for every $i<\omega$ and we
can apply 3-amalgamation as before to find $A^{\prime\prime}$ such that
$A^{\prime\prime}B_{i}\equiv_{E}AB_{0}$ for all $i<\omega$.∎

###  2.2 The parameterized class

####  2.2.1 The base case

In this section we consider a Fraïssé class $K_{0}$ and a limit $T_{0}$
satisfying the conditions $(H)$, we show that we can ’parameterize’ this class
to get a new class $K$ of structures in a new language that also satisfies
$(H)$, and that in some cases the structure we get by constructing a limit of
$K$ is not simple, and so strictly NSOP1 by Corollary 11. We need the
assumption that any of the structures of $K_{0}$ can be embedded together, or
equivalently that the structures of $K_{0}$ agree on what the isomorphism type
of structure generated by the constants is. This can be done by restricting
$K_{0}$ to a smaller class.

Let $L_{0}$ be a countable language and $K_{0}$ a Fraïssé class of finite
$L_{0}$-structures satisfying $(H)$, so :

  1. 1.

(H.P.) : If $A\subseteq B\in K_{0}$ is a substructure then $A\in K_{0}$

  2. 2.

(Free Amalgamation) : If $C\subseteq A,B$ are substructures, there is $D\in
K_{0}$ and embeddings $i_{A},i_{B}$ of $A$ and $B$ respectively to $D$
satisfying $i_{A}|_{C}=i_{B}|_{C}$, $i_{A}(A)\cap i_{B}(B)=i_{A}(C)$, $D$ is
generated by the images of $A$ and $B$ and such that given any morphism of
$L$-structures $\varphi_{A}:A\rightarrow D^{\prime}$,
$\varphi_{B}:B\rightarrow D^{\prime}$ to $D^{\prime}\in K_{0}$ there is a
unique morphism $\varphi:D\rightarrow D^{\prime}$ such that
$\varphi_{A}=\varphi\cdot i_{A}$ and $\varphi_{B}=\varphi\cdot i_{B}$. We will
write $A\oplus_{C}B$ for this structure $D$.

  3. 3.

(Generic extension) : If $A\in K_{0}$ and $V$ is a sort there is an extension
$A\oplus_{\emptyset}\langle x\rangle\in K_{0}$ generated by a single element
$x\in V$ over $A$ such that for every morphism $\varphi:A\rightarrow
A^{\prime}\in K_{0}$ and every element $a^{\prime}\in V(A^{\prime})$ there is
a unique morphism $\psi:A\oplus_{\emptyset}\langle x\rangle\rightarrow
A^{\prime}$ extending $\varphi$ by sending $x$ to $a^{\prime}$.

  4. 4.

$K_{0}$ is axiomatisable : if the isomorphism class of an $L$-structure $A$ is
not inside of $K$ there is a formula $\varphi$ and a finite tuple $a\in A$
such that $A\models\varphi(a)$ and
$A^{\prime}\centernot\models\varphi(a^{\prime})$ for every $a^{\prime}\in
A^{\prime}\in K$.

  5. 5.

Any limit of $K_{0}$ has Q.E. : for every quantifier free formula
$\varphi(x,y)$ satisfied in some structure of $K_{0}$ there is a quantifier
free formula $\psi(x)$ such that if $a\in A\in K_{0}$ and $A\models\psi(a)$
then there is $A\subseteq B\in K_{0}$ an extension and $b\in B$ such that
$B\models\varphi(a,b)$.

  6. 6.

($3$ Amalgamation for algebraically independent extension). Given a
commutative diagram in $K_{0}$ as on the left, where the arrows are
embeddings, $\varphi_{B_{i}}(B_{i})\cap\varphi_{A_{i}}(A)=\varphi_{A_{i}}\cdot
i_{A}(E)=\varphi_{B_{i}}\cdot i_{B_{i}}(E)$ for $i=0,1$ and
$\psi_{B_{0}}(B_{0})\cap\psi_{B_{1}}(B_{1})=\psi_{B_{0}}\cdot
i_{B_{0}}(E)=\psi_{B_{1}}\cdot i_{B_{1}}(E)$, we can complete it inside of
$K_{0}$ into a commutative diagram as on the right (where the arrows are
embeddings) such that
$\varphi_{D_{i}}(D_{i})\cap\varphi_{B}(B)=\varphi_{D_{i}}\cdot\varphi_{B_{i}}(B_{i})=\varphi_{B}\cdot\psi_{B_{i}}(B_{i})$
for $i=0,1$ and
$\varphi_{D_{0}}(D_{0})\cap\varphi_{D_{1}}(D_{1})=\varphi_{D_{0}}\cdot\varphi_{A_{0}}(A)=\varphi_{D_{1}}\cdot\varphi_{A_{1}}(A)$.

${{D_{1}}}$${{D_{1}}}$${D}$${A}$${{D_{0}}}$${A}$${{D_{0}}}$${{B_{1}}}$${B}$${{B_{1}}}$${B}$${E}$${{B_{0}}}$${E}$${{B_{0}}}$$\scriptstyle{i_{B_{0}}}$$\scriptstyle{\psi_{B_{0}}}$$\scriptstyle{i_{B_{1}}}$$\scriptstyle{i_{A}}$$\scriptstyle{\varphi_{A_{0}}}$$\scriptstyle{\varphi_{B_{0}}}$$\scriptstyle{\varphi_{A_{1}}}$$\scriptstyle{\psi_{B_{1}}}$$\scriptstyle{\varphi_{D_{0}}}$$\scriptstyle{\varphi_{B}}$$\scriptstyle{\varphi_{D_{1}}}$

Write $T_{0}$ the theory of some fixed limit of $K_{0}$ and let
$\mathbb{M}_{0}$ be a monster model of $T_{0}$. By the previous section we
know that $T_{0}$ has Q.E., satisfies that $acl(A)=dcl(A)=\langle A\rangle$
and that $\mathrel{\mathop{\vbox{
\hbox{\oalign{\kern-1.29167pt\cr\hfil$|$\hfil\cr\kern-3.01385pt\cr$\smile$\cr\kern-1.29167pt\cr}}
}}\displaylimits}^{a}$ satisfies the independence theorem over arbitrary sets,
which implies that $T_{0}$ is NSOP1 and that $\mathrel{\mathop{\vbox{
\hbox{\oalign{\kern-1.29167pt\cr\hfil$|$\hfil\cr\kern-3.01385pt\cr$\smile$\cr\kern-1.29167pt\cr}}
}}\displaylimits}^{K}=\mathrel{\mathop{\vbox{
\hbox{\oalign{\kern-1.29167pt\cr\hfil$|$\hfil\cr\kern-3.01385pt\cr$\smile$\cr\kern-1.29167pt\cr}}
}}\displaylimits}^{a}$ over arbitrary sets in $\mathbb{M}_{0}$.

######  Definition 1.

Let $L$ be a new language built from $L_{0}$ by adding a new sort $P$ (we will
refer to the sorts of $L_{0}$ as the object sorts and write them $O$) and
replacing every symbol of constant $c$ in $L_{0}$ of sort $O$ by a function
symbol $c_{\mathunderscore}:P\rightarrow O$, every function symbol
$f:\prod_{i<n}O_{i}\rightarrow\prod_{i<m}O^{\prime}_{i}$ by and function
symbol
$f_{\mathunderscore}:P\times\prod_{i<n}O_{i}\rightarrow\prod_{i<m}O^{\prime}_{i}$
and similarly for relation symbols. For an $L$ structure $A$ and $p\in P(A)$
we write $A_{p}$ to denote the $L_{0}$-structure induced by $p$ on $O(A)$. For
a set of objects $A$ and a tuple of parameters $\overline{p}$ we will write
$\langle A\rangle_{\overline{p}}$ for the structure generated by
$(A,\overline{p})$, we will also write $\langle A\rangle$ for the structure
generated by a set $A$ (eventually containing parameters and objects), so
$\langle A\rangle=\langle O(A)\rangle_{P(A)}$.

######  Definition 2.

Let $K$ be the class of countable $L$-structures $A$ such that $A_{p}\in
K_{0}$ for every $p\in P(A)$.

Let us set some notations : we will write
$A\oplus_{\emptyset}\langle\overline{x}\rangle$ for the structure freely
generated over $A\subseteq\mathbb{M}_{0}$ by a tuple $\overline{x}$ of
distinct elements. Such an extension exists by the (Generic extension)
assumption. For an $L_{0}$-formula $\varphi$ and a parameter $p$ we will write
$\varphi_{p}$ for the $L$-formula parameterized by $p$.

######  Remark 3.

The class $K$ is axiomatisable : if an $L$-structure $A$ is not in $K$ then
for some parameter $p\in P(A)$ there is some $p\in A$ such that
$A_{p}\centernot\in K_{0}$, so there is an $L_{0}$-formula $\varphi$ such that
$A_{p}\models\varphi$ and such that no structures in $K_{0}$ satisfy
$\varphi$. So $A\models\varphi_{q}$ and no structure in $K$ satisfy this
formula.

######  Lemma 4.

The class $K$ satisfies the condition of (Free Amalgamation).

###### Proof.

Consider the structures $E\subseteq A,B\in K$. We will construct an
$L$-structure in $K$ satisfying the (Free Amalgamation) universal property.
Let us write $E=(O(E),P(E))$, $A=(O_{A},P_{A})$ and $B=(O_{B},P_{B})$. We
begin by defining $P(D):=(P(A)\setminus P(E))\dot{\cup}(P(B)\setminus P(E))$
which will be the parameter sort of the free amalgam in $K$. Let
$(p_{i})_{i<\omega}$ enumerate $P(D)$. We define a sequence
$((D^{k}_{i})_{i<\omega})_{k<\omega}$ of tuples of countable
$L_{0}$-structures indexed by $P_{D}$ by induction. Let $D^{0}_{-1}:=A\cup B$.

We define
$D^{0}_{p_{i}}:=(A_{p_{i}}\oplus_{E_{p_{i}}}B_{p_{i}})\oplus_{\emptyset}\langle
D^{0}_{p_{i-1}}\setminus(A\cup B)\rangle$ if $p_{i}\in P(E)$,
$D^{0}_{p_{i}}:=A_{p_{i}}\oplus_{\emptyset}\langle D^{0}_{p_{i-1}}\setminus
A\rangle$ if $p_{i}\in P(A)\setminus P(E)$,
$D^{0}_{p_{i}}:=B_{p_{i}}\oplus_{\emptyset}\langle D^{0}_{p_{i-1}}\setminus
B\rangle$ if $p_{i}\in P(B)\setminus P(E)$, and finally
$D^{0}_{p_{i}}:=E_{0}\oplus_{\emptyset}\langle D^{0}_{p_{i-1}}\rangle$ if
$p_{i}\in P(D)\setminus(P(A)\cup P(B))$. Then set
$D^{0}=\bigcup\limits_{i<\omega}D^{0}_{p_{i}}$. If $D^{k}$ is defined for some
$1\leq k<\omega$ we set
$D^{k+1}_{p_{0}}=D^{k}_{p_{0}}\oplus_{\emptyset}\langle D^{k}\setminus
D^{k}_{p_{0}}\rangle$, similarly if $D^{k}_{p_{i}}$ is defined for some $1\leq
i<\omega$ we set $D^{k+1}_{p_{i+1}}=D^{k}_{p_{i+1}}\oplus_{\emptyset}\langle
D^{k+1}_{i}\setminus D^{k}_{p_{i}}\rangle$.

Every $D^{k}_{p}$ is an $L_{0}$-structure for the parameter $p$. We set
$O(D)=\bigcup\limits_{k<\omega}D^{k}$, this set is countable and has an
$L_{0}$-structure for the parameter $p$ given by
$D_{p}=\bigcup\limits_{k<\omega}D^{k}_{p}\in K_{0}$, so $D\in K$. This defines
an $L$ structure $D=(O(D),P(D))\in K$ which extends $A$ and $B$, and such that
$A\cap B=E$ inside of $D$. Also if $A$ and $B$ are generated by
$(\overline{a},P(A))$ and $(\overline{b},P(B))$ respectively then
$(\overline{ab},P(A),P(B))$ generates $D$.

Now that we constructed the structure $D$ we show that is satisfies the right
property. Consider $D^{\prime}\in K$, $\varphi_{A}:A\rightarrow D^{\prime}$
and $\varphi_{B}:B\rightarrow D^{\prime}$ as in the statement, and write
$\varphi_{E}$ their restriction to $E$. The uniqueness of $\varphi$ is clear
from the fact that $D$ is generated by the images of $A$ and $B$. To prove
existence we construct an increasing sequence of tuples of morphism of
$L_{0}$-structures indexed on $P(D)$ :
$((\varphi^{k}_{p_{i}}:D^{k}_{p_{i}}\rightarrow
D^{\prime}_{\varphi_{P}(p_{i})})_{i<\omega})_{k<\omega}$. Let us begin by
defining $\varphi$ on the parameters, let $\varphi(p_{i})=\varphi_{E}(p_{i})$
if $p_{i}\in P(E)$, $\varphi(p_{i})=\varphi_{A}(p_{i})$ if $p_{i}\in
P(A)\setminus P(E)$ and $\varphi(p_{i})=\varphi_{B}(p_{i})$ if $p_{i}\in
P(B)\setminus P(E)$.

For $k=0$, if $p_{i}\in P(E)$ we take the unique morphism
$\varphi^{0}_{p_{i}}:D^{0}_{p_{i}}\rightarrow D^{\prime}_{\varphi_{E}(p_{i})}$
that extends $\varphi_{A}(p,\mathunderscore):A_{p}\rightarrow
D^{\prime}_{\varphi_{E}(p_{i})}$ and
$\varphi_{B}(p,\mathunderscore):B_{p}\rightarrow
D^{\prime}_{\varphi_{E}(p_{i})}$ and sends any element $d\in D^{0}_{p_{i-1}}$
to $\varphi^{0}_{p_{i-1}}(d)$. If $p_{i}\in P(A)\setminus P(E)$ we take the
unique morphism $\varphi^{0}_{p_{i}}:D^{0}_{p_{i}}\rightarrow
D^{\prime}_{\varphi_{A}(p_{i})}$ that extends
$\varphi_{A}(p,\mathunderscore):A_{p}\rightarrow
D^{\prime}_{\varphi_{A}(p_{i})}$ and sends any element $d\in D^{0}_{p_{i-1}}$
to $\varphi^{0}_{p_{i-1}}(d)$, similarly if $p_{i}\in P(B)\setminus P(E)$ we
take the unique morphism $\varphi^{0}_{p_{i}}:D^{0}_{p_{i}}\rightarrow
D^{\prime}_{\varphi_{B}(p_{i})}$ that extends
$\varphi_{B}(p,\mathunderscore):B_{p_{i}}\rightarrow
D^{\prime}_{\varphi_{B}(p_{i})}$ and sends any element $d\in D^{0}_{p_{i-1}}$
to $\varphi^{0}_{p_{i-1}}(d)$.

Assume that for every $i<\omega$ the functions
$\varphi^{k}_{p}:D^{k}_{p}\rightarrow D^{\prime}_{\varphi(p)}$ have been
defined. Define $\varphi^{k+1}_{p_{0}}:D^{k+1}_{p_{0}}\rightarrow D^{\prime}$
as the unique morphism extending $\varphi^{k}_{p_{0}}:D^{k}_{p}\rightarrow
D^{\prime}$ by sending $x\in D^{k}_{p_{i}}$ to $\varphi^{k}_{p_{i}}(x)$ for
every $i<\omega$. We just repeat the same steps for the other parameters. Once
all these functions are constructed we consider
$\varphi=\bigcup\limits_{k<\omega}\varphi^{k}_{p}$ for every $p\in P(D)$. This
function defines a map in $K$ from $D$ to $D^{\prime}$ such that its
restrictions to $A$ and $B$ are $\varphi_{A}$ and $\varphi_{B}$ respectively,
and it is a morphism of $L_{0}$-structure since
$\varphi=\bigcup\limits_{k<\omega}\varphi^{k}_{p}:D_{p}\rightarrow
D^{\prime}_{\varphi(p)}$ is a morphism of $L_{0}$-structure for every
parameter $p\in P(D)$.∎

######  Remark 5.

This means that $K$ seen as a category has pushouts for pairs of
monomorphisms. The proof also implies that the construction of $D$ does not
depend (up to isomorphism) on the choices of the bases of $A$ and $B$ and on
the enumeration of the parameters : given the same construction for a
different enumeration $D^{*}=\bigcup\limits_{i<\omega}D^{*,k}_{p}$ we can show
that $D^{*}$ also has this universal property with a similar proof.

######  Remark 6.

From the proof of Lemma 4 it is clear how to show that $K$ satisfies the
(Generic extension property), we begin with a sort among the object sorts, we
fix some enumeration $(p_{i})_{i<\omega}$ of the parameters, we begin by
adding one object $x$ in the right sort and consider the associated
$L_{0}$-structure $A^{0}_{p_{0}}=A\oplus_{\emptyset}\langle x\rangle$, then
$A^{0}_{p_{i+1}}=A\oplus_{\emptyset}\langle A^{0}_{p_{i}}\setminus A\rangle$,
$A^{k+1}_{p_{i+1}}=A^{k}_{p_{i+1}}\oplus_{\emptyset}\langle
A^{k+1}_{p_{i}}\setminus A^{k}_{p_{i+1}}\rangle$ and so on. When adding a
generic parameter $p$ we consider the $L_{0}$-structure $E_{0}$ it generates,
i.e. the structure generated by the constants, and form
$A\oplus_{\emptyset}\langle E_{0},p\rangle$ the same way but with adding a
parameter $p$ before the enumeration $(p_{i})_{i<\omega}$ of $P(A)$ for which
we consider the structure freely generated by $A$ over $E_{0}$,
$E_{0}\oplus_{\emptyset}\langle A\rangle$. We can indeed send $p$ to any
parameter because any parameter agree with $p$ on the structure generated by
the constants.

A substructure of a finitely generated structure is not necessarily finitely
generated : an $L_{P}$ structure with only one parameter is the same as an
$L_{0}$-structure, so depending on $T_{0}$ (if it is uniformly locally finite
for example) by removing all parameters except one we can find examples of
this fact. With the properties of $K$ we have proven so far we can construct a
limit $\mathcal{M}$ with Proposition 2 and consider its theory $T$ and
$\mathbb{M}$ a monster model of it.

######  Lemma 7.

$K$ satisfies $4$ : for every quantifier free formula $\varphi(x,y)$ satisfied
in some structure of $K$ there is a quantifier free formula $\psi(x)$ such
that if $a\in A\in K$ and $A\models\psi(a)$ then there is $A\subseteq B\in K$
an extension and $b\in B$ such that $B\models\varphi(a,b)$.

Such a formula $\varphi(x,y)$ can be decomposed the following way : take
$x=x^{\prime},\overline{p}$, $y=y^{\prime}\overline{q}$ with
$x^{\prime},y^{\prime}$ in the object sorts and $\overline{p},\overline{q}$ in
the parameters, then $\varphi(x,y)$ is implied by a formula of the form
$\bigwedge\limits_{i<n}\varphi_{i,p_{i}}(x^{\prime},y^{\prime})\wedge\bigwedge\limits_{j<m}\varphi^{\prime}_{j,q_{j}}(x^{\prime},y^{\prime})$
where the $\varphi_{i},\varphi^{\prime}_{j}$ are quantifier free
$L_{0}$-formulas and the parameters $p_{i}$ and $q_{j}$ are all distinct. By
assumption if $\psi_{i}$ is the formula associated to
$\varphi_{i}(x^{\prime},y^{\prime})$ then the formula
$\bigwedge\limits_{i<n}\psi_{i,p_{i}}(x^{\prime})\wedge\bigwedge\limits_{i\centernot=i^{\prime}<n}(p_{i}\centernot=p_{j})$
satisfies our condition : if $(a^{\prime},\overline{p}^{\prime})\in
A^{\prime}\in K$ satisfies it by assumption there are $L_{0}$-structures
$(B^{\prime}_{i})_{i<n}$ and $b^{\prime}_{i}\in B^{\prime}_{i}$ such that
$A^{\prime}_{p_{i}}\subseteq
B^{\prime}_{i}\models\varphi_{i}(a^{\prime},b_{i}^{\prime})$. For $j<m$ we can
find some $b^{\prime}_{q_{j}}\in B_{q_{j}}\in K_{0}$ containing the objects of
$A^{\prime}$ such that
$B_{q_{j}}\models\varphi^{\prime}_{j}(a^{\prime},b_{q_{j}}^{\prime})$. We can
build $B^{\prime}$ an $L$ structure extending $A^{\prime}$ with parameters
$P(A^{\prime})\cup\\{q_{j}$ : $j<m\\}$ by amalgamating all of these
$L_{0}$-structure freely over $A\cup b^{\prime}$ (as in the proof of Lemma 4),
we then have that $B^{\prime}\models\varphi(a^{\prime},b^{\prime})$. Once this
condition is satisfied we can show with the same proof as in Lemma 6 that $T$
has Q.E. in $L$.

######  Proposition 8.

$T$ proves the following statements :

  1. 1.

If $\overline{a}$ is a finite tuple of objects and $\pi_{i}(\overline{x})$ is
a finitary type in $T_{0}$ for every $i\leq n$ such that $\pi_{i}\vdash
x_{j}\centernot=x_{j^{\prime}}$ for $j\centernot=j^{\prime}$ there exist an
infinite number of tuples of parameters $(p_{i})_{i\leq n}$ such that
$\models\pi_{i}(\overline{a}_{p_{i}})$ for every $i\leq n$.

  2. 2.

If $\overline{p}=p_{0},..,p_{n}$ is a finite tuple of parameters and
$\pi_{i}(\overline{x},\overline{b}_{p_{i}})$ is a type in
$\mathcal{M}_{p_{i}}$ over a tuple of objects $\overline{b}$ such that the
induced type in any of the $x_{i}$ is non algebraic and that $\pi_{i}\vdash
x_{j}\centernot=x_{j^{\prime}}$ for $j\centernot=j^{\prime}$ then there is a
tuple of objects $\overline{a}$ such that
$\models\pi_{i}(\overline{a}_{p_{i}},\overline{b}_{p_{i}})$ for every $i\leq
n$.

###### Proof.

Consider the structure $\mathcal{M}$ from Proposition 2, a finite tuple of
object $\overline{a}\in\mathcal{M}$ and a finite set of types
$\pi_{i}(\overline{x})$ as in the first point of the statement. We can build a
structure $A\in K$ containing $\overline{a}$ and with parameters
$(p_{i})_{i\leq n}$ such that $\models\pi_{i}(\overline{a}_{p_{i}})$ for every
$i\leq n$. Then by saturation we can embed this structure $A$ inside of
$\mathcal{M}$ over $\overline{a}$ and find the parameters. The second point is
similar : consider the structure $B$ generated by $\overline{b},\overline{p}$
and types $\pi_{i}(\overline{x},\overline{b}_{p_{i}})$ as in the statement. We
can extend $B$ into $B^{\prime}\in K$ that contains a tuple of objects
$\overline{a}^{\prime}$ such that $\overline{a}^{\prime}\overline{b}$ has the
right type for the parameter $p_{i}$ for every $i$ and then embed $B^{\prime}$
into $\mathcal{M}$ over $B$ to find the right objects.∎

######  Corollary 9.

The theory $T$ has I.P. if there is two distinct non algebraic types in
$T_{0}$ and is non-simple if the generated structure in $K_{0}$ is non
trivial, in the sense that we can find $c\in\langle ab\rangle\setminus(\langle
a\rangle\cup\langle b\rangle)$ for some $a,b\in\mathbb{M}_{0}$.

###### Proof.

For I.P. consider two distinct $T$-types $\pi_{i}$ in $\overline{x}$ for
$i=0,1$. Now if we consider a sequence $(p_{i})_{i<\omega}$ of distinct
parameters in $\mathbb{M}$ then by 8 for every $I\subseteq\omega$ we can find
$\overline{a}_{I}$ such that $\models\pi_{0}(\overline{a}_{I})$ if and only if
$i\in I$.

Now for simplicity let $t(x,y)$ be a $L_{0}$-term such that
$t(a,b)\centernot\in\langle a\rangle\cup\langle b\rangle$ and consider a tree
of parameters all distinct $(p_{\mu})_{\nu\in\omega^{<\omega}}$. Lets now
consider a tree of distinct objects $(a_{\mu})_{\mu\in\omega^{<\omega}}$ and
the formulas $\varphi(x,y,a_{\mu\frown
i},p_{\mu}):=t_{p_{\mu}}(x,y)=a_{\mu\frown i}$ along this tree. Clearly
$\\{\varphi(x,y,a_{\mu\frown i},p_{\mu})$ : $i<\omega\\}$ is 2-inconsistent
for every $\mu\in\omega^{<\omega}$, and by 8 every path is consistent. ∎

######  Remark 10.

The relation $\mathrel{\mathop{\vbox{
\hbox{\oalign{\kern-1.29167pt\cr\hfil$|$\hfil\cr\kern-3.01385pt\cr$\smile$\cr\kern-1.29167pt\cr}}
}}\displaylimits}^{\Gamma}$ in $\mathbb{M}$ does not necessarily satisfy local
character (otherwise it would satisfy all conditions of the Kim-Pillay
theorem) : take some distinct objects $\overline{o}=o_{1},o_{2},o_{3}$ and a
set $P=(p_{i})_{i<\kappa}$ of parameters for $\kappa=|T_{P}|^{+}$ such that
$o_{3}\in\langle o_{1}o_{2}\rangle_{p_{i}}$ for every $i<\kappa$ and
$o_{j}\notin\langle P\rangle$ for every $j\in\\{0,1,2\\}$. Then there is no
subset $P_{0}$ of cardinal $|T|$ of $P$ such that
$\overline{o}\mathrel{\mathop{\vbox{
\hbox{\oalign{\kern-1.29167pt\cr\hfil$|$\hfil\cr\kern-3.01385pt\cr$\smile$\cr\kern-1.29167pt\cr}}
}}\displaylimits}^{\Gamma}_{P_{0}}P$.

######  Remark 11.

In the case where we can find in $K_{0}$ some $c=t(a,b)\in\langle
ab\rangle\setminus(\langle a\rangle\cup\langle b\rangle)$ then the relation
$\mathrel{\mathop{\vbox{
\hbox{\oalign{\kern-1.29167pt\cr\hfil$|$\hfil\cr\kern-3.01385pt\cr$\smile$\cr\kern-1.29167pt\cr}}
}}\displaylimits}^{a}$ in $\mathbb{M}$ does not admit $4$-amalgamation : take
3 distinct parameters $p_{0},p_{1},p_{2}$ and distinct objects
$b_{0},b_{1},b_{2}$ such that $b_{i},b_{j}\centernot\in\langle p_{k}\rangle$
for every $i,j,k$ such that $\\{i,j,k\\}=\\{0,1,2\\}$ and such that
$t_{p_{0}}(b_{1},b_{2})=t_{p_{1}}(b_{0},b_{2})$,
$t_{p_{1}}(b_{0},b_{2})=t_{p_{2}}(b_{0},b_{1})$ and
$t_{p_{0}}(b_{1},b_{2})\centernot=t_{p_{2}}(b_{1},b_{2})$. Then the type
$\mbox{\rm tp}(p_{0}/b_{1}b_{2})\cup\mbox{\rm
tp}(p_{1}/b_{0}b_{2})\cup\mbox{\rm tp}(p_{2}/b_{0}b_{1})$ is inconsistent.

Since the theory $T_{0}$ has Q.E., is ’one based for $\mathrel{\mathop{\vbox{
\hbox{\oalign{\kern-1.29167pt\cr\hfil$|$\hfil\cr\kern-3.01385pt\cr$\smile$\cr\kern-1.29167pt\cr}}
}}\displaylimits}^{K}$’ and since the algebraic closure and the generated
structure coincide in $\mathbb{M}_{0}$ we can represent non-Kim-forking
extension as commutative squares in the following way : the arrows are
embeddings, the structure $D$ is generated by the images $f(A)$ and $g(B)$,
and $f(A)\cap g(B)=f\cdot j(E)=g\cdot i(E)$. Then the isomorphism type of $D$
over $B$ gives us a type over $D$ which correspond to a non-forking extension
of $\mbox{\rm tp}(A/E)$. We will refer to such squares as strong amalgams.
However for the ($3$ Amalgamation for algebraically independent extension) the
squares we are considering consists in an extension of such a square, meaning
that we compose with an embedding from $D$ to some $D^{\prime}\in K_{0}$.

${A}$${D}$${E}$${B}$$\scriptstyle{i}$$\scriptstyle{g}$$\scriptstyle{j}$$\scriptstyle{f}$

######  Proposition 12.

$K$ satisfies ($3$ Amalgamation for algebraically independent extension).

###### Proof.

Consider the following diagram in $K$, which is as in the definition of ($3$
Amalgamation for algebraically independent extension) and where for simplicity
we take the embeddings to be inclusions. We want to complete it, for this we
begin by choosing a completion for every parameter. Fix an enumeration
$(p_{i})_{i<\omega}$ of $P(D):=P(D_{0})\dot{\cup}[P(D_{1})\setminus
P(A)]\dot{\cup}[P(B)\setminus P(B_{0})\dot{\cup}P(B_{1}))]$, we will take this
set as the set of parameters of our structure $D$. Looking at the
$L_{0}$-structure indexed by any of the parameters $p_{j}\in P(E)$ we can find
an $L_{0}$-structure $D^{*,0}_{p_{j}}$ as in the right diagram.

${{D_{1}}}$${{D_{1,p}}}$${D^{*,0}_{p_{j}}}$${A}$${{D_{0}}}$${A_{p_{j}}}$${{D_{0,p_{j}}}}$${{B_{1}}}$${B}$${{B_{1,p_{j}}}}$${B_{p_{j}}}$${E}$${{B_{0}}}$${E_{p_{j}}}$${{B_{0,p_{j}}}}$$\scriptstyle{i_{B_{0}}}$$\scriptstyle{\psi_{B_{0}}}$$\scriptstyle{i_{B_{1}}}$$\scriptstyle{i_{A}}$$\scriptstyle{\varphi_{A_{0}}}$$\scriptstyle{\varphi_{B_{0}}}$$\scriptstyle{\varphi_{A_{1}}}$$\scriptstyle{\psi_{B_{1}}}$$\scriptstyle{\varphi_{D_{0}}}$$\scriptstyle{\varphi_{B}}$$\scriptstyle{\varphi_{D_{1}}}$

These sets $D^{*,0}_{p_{j}}$ for $p_{j}\in P(E)$ are also equipped with the
$L$-structure on the sets $\varphi_{B}(B)$ and $\varphi_{D_{i}}(D_{i})$ for
$i=0,1$ induced by the embeddings. We now define their equivalent for the
parameters of $D_{i}$ and $B$. Let $p_{j}\in P(A)\setminus P(E)$. We consider
the $L_{0}$-structure
$D^{*,0}_{p_{j}}=[D_{0,p_{j}}\oplus_{A_{p_{j}}}D_{1,p_{j}}]\oplus_{\emptyset}\langle
B\setminus(B_{0}\cup B_{1})\rangle$ with the natural mappings from $D_{0}$,
$D_{1}$ and $B$ to $D^{*,0}_{p_{j}}$. Similarly if $p_{j}\in P(B_{i})\setminus
P(E)$ set $D^{*,0}_{p_{j}}=[D_{0}\oplus_{B_{0}}B]\oplus_{\emptyset}\langle
D_{1}\setminus(A\cup B_{1})\rangle$ with the natural mappings from $D_{0}$,
$D_{1}$ and $B$ to $D^{*,0}_{p_{j}}$. If $p_{j}\in P(D_{0})\setminus(P(A)\cup
P(B_{0}))$, set
$D^{*,0}_{p_{j}}=D_{0,p_{j}}\oplus_{\emptyset}\langle(B\setminus
B_{0})\cup(D_{1}\setminus A)\rangle$ with the natural mappings from $D_{0}$,
$D_{1}$ and $B$ to $D^{*,0}_{p_{j}}$ and similarly for $p_{j}\in
P(D_{1})\setminus(P(A)\cup P(B_{1}))$. If $p_{j}\in P(B)\setminus(P(B_{0})\cup
P(B_{1}))$, set
$D^{*,0}_{p_{j}}=B_{p_{j}}\oplus_{\emptyset}\langle(D_{0}\setminus
B_{0})\cup(D_{1}\setminus B_{1})\rangle$ with the natural mappings from
$D_{0}$, $D_{1}$ and $B$ to $D^{*,0}_{p_{j}}$.

Now that we have all of our $L_{0}$-structures we will just put them together
in a canonical way : we define an increasing sequence of tuples of
$L_{0}$-structures $((D^{k}_{p_{j}})_{j<\omega})_{k<\omega}$. Let
$D^{0}_{p_{0}}=D^{*,0}_{p_{0}}$, if $D^{0}_{p_{j}}$ has been defined for some
$j<\omega$ let $D^{0}_{p_{j+1}}=D^{*,0}_{p_{j+1}}\oplus_{\emptyset}\langle
D^{0}_{p_{j}}\setminus(D_{0}\cup D_{1}\cup B\rangle)\rangle$ where we
assimilate $D_{0}$, $D_{1}$ and $B$ with their image in $D^{0}_{p_{j}}$. Once
$D^{0}_{p_{j}}$ has been defined for every $j<\omega$ we set
$D^{0}=\bigcup\limits_{i<\omega}D^{0}_{p_{j}}$. If $D^{k}$ is defined we let
$D^{k+1}_{p_{0}}=D^{k}_{p_{0}}\oplus_{\emptyset}\langle D^{k}\setminus
D^{k}_{p_{0}}\rangle$. Now assume that $D^{k+1}_{p_{j}}$ is defined for some
$k<\omega$,$j<\omega$, we set
$D^{k+1}_{p_{j+1}}=D^{k}_{p_{j+1}}\oplus_{\emptyset}\langle
D^{k+1}_{p_{j}}\setminus D^{k}_{p_{j+1}}\rangle$. Then take
$D=\bigcup\limits_{k<\omega}D^{k}_{p_{j}}$ with the $L_{0}$-structure induced
for every $j<\omega$, this defines an $L$-structure, the natural mappings we
had from $D_{0}$, $D_{1}$ and $B$ to $D$ are embeddings for the $L$-structure
we put on $D$ and they satisfy the conditions.∎

Finally we can state that the class $K$ satisfies the conditions of $(H)$, and
so that $T$ is an NSOP1 theory with existence such that
$\mathrel{\mathop{\vbox{
\hbox{\oalign{\kern-1.29167pt\cr\hfil$|$\hfil\cr\kern-3.01385pt\cr$\smile$\cr\kern-1.29167pt\cr}}
}}\displaylimits}^{K}=\mathrel{\mathop{\vbox{
\hbox{\oalign{\kern-1.29167pt\cr\hfil$|$\hfil\cr\kern-3.01385pt\cr$\smile$\cr\kern-1.29167pt\cr}}
}}\displaylimits}^{a}$ over arbitrary sets.

####  2.2.2 Some structure on the parameters

In this subsection we describe how we can generalise the previous part by
having some structure on the parameters, the proof are essentially similar so
we wont give as much details. Let $K_{0}$ be a class of $L_{0}$ structures as
in the previous section : it satisfies $(H)$ and there is only one isomorphism
type for the structure generated by the constants. Let us consider an other
class $K_{P}$ of one-sorted $L_{P}$-structures satisfying $(H)$.

######  Definition 1.

Let $L$ be a new language built from $L_{0}$ and $L_{P}$ by adding a new sort
$P$ (we will refer to the sorts of $L_{0}$ as the object sorts and write them
$O$) to $L_{0}$ which we equip with $L_{P}$, and replacing every symbol of
constant $c$ in $L_{0}$ of sort $O$ by a function symbol
$c_{\mathunderscore}:P\rightarrow O$, every function symbol
$f:\prod_{i<n}O_{i}\rightarrow\prod_{i<m}O^{\prime}_{i}$ by and function
symbol
$f_{\mathunderscore}:P\times\prod_{i<n}O_{i}\rightarrow\prod_{i<m}O^{\prime}_{i}$
and similarly for relation symbols. For an $L$ structure $A$ and $p\in P(A)$
we write $A_{p}$ to denote the $L_{0}$-structure induced by $p$ on $O(A)$. For
a set of objects $A$ and a tuple of parameters $\overline{p}$ we will write
$\langle A\rangle_{\overline{p}}$ for the structure generated by
$(A,\overline{p})$, we will also write $\langle A\rangle$ for the structure
generated by a set $A$ (eventually containing parameters and objects), so
$\langle A\rangle=\langle O(A)\rangle_{P(A)}$.

######  Definition 2.

Let $K$ be the class of countable $L$-structures $A$ such that $P(A)\in K_{P}$
and $A_{p}\in K_{0}$ for every $p\in P(A)$. In particular the previous section
corresponds to the case of $K_{P}$ being the class of countable sets with the
equality.

######  Remark 3.

The class $K$ is axiomatisable : if an $L$-structure $A$ is not in $K$ then
either for some parameter $p\in P(A)$ there is some $p\in A$ such that
$A_{p}\centernot\in K_{0}$, in which case the same argument as in Remark 3
applies, or $P(A)\centernot\in K_{P}$, which is then also expressed by a
formula by $4$ in $K_{P}$. Also $K$ clearly satisfies $(H.P.)$.

######  Lemma 4.

$K$ satisfies $4$ : for every quantifier free formula $\varphi(x,y)$ satisfied
in some structure of $K$ there is a quantifier free formula $\psi(x)$ such
that if $a\in A\in K$ and $A\models\psi(a)$ then there is $A\subseteq B\in K$
an extension and $b\in B$ such that $B\models\varphi(a,b)$.

###### Proof.

Such a formula $\varphi(x,y)$ can be decomposed the following way : take
$x=x^{\prime},\overline{p}$, $y=y^{\prime}\overline{q}$ with
$x^{\prime},y^{\prime}$ in the object sorts and $\overline{p},\overline{q}$ in
the parameters, then $\varphi(x,y)$ is implied by a formula of the form
$\varphi(\overline{p},\overline{q})\wedge\bigwedge\limits_{i<n}\varphi_{i,p_{i}}(x^{\prime},y^{\prime})\wedge\bigwedge\limits_{j<m}\varphi^{\prime}_{j,q_{j}}(x^{\prime},y^{\prime})$
where $\varphi$ a quantifier free $L_{P}$-formula, the
$\varphi_{i},\varphi^{\prime}_{j}$ are quantifier free $L_{0}$-formulas and
the parameters $p_{i}$ and $q_{j}$ are all distinct. By assumption if $\psi$
is the formula associated to $\varphi$ by $5$ in $K_{P}$ and $\psi_{i}$ is the
formula associated to $\varphi_{i}(x^{\prime},y^{\prime})$ by $5$ in $K_{0}$
then the formula
$\psi(\overline{p})\wedge\bigwedge\limits_{i<n}\psi_{i,p_{i}}(x^{\prime})$
satisfies our condition : if $(a^{\prime},\overline{p}^{\prime})\in
A^{\prime}\in K$ satisfies it by assumption there is $\overline{q}\in
B_{P}^{\prime}\supseteq P(A^{\prime})$ an $L_{P}$ structure such that
$B_{P}^{\prime}\models\varphi(\overline{p},\overline{q})$. We also have
$L_{0}$-structures $(B^{\prime}_{i})_{i<n}$ and $b^{\prime}_{i}\in
B^{\prime}_{i}$ such that $A^{\prime}_{p_{i}}\subseteq
B^{\prime}_{i}\models\varphi_{i}(a^{\prime},b_{i}^{\prime})$. Now we construct
some extension $B^{\prime}$ of $A^{\prime}$ in $K$ from all of these pieces.
We set $P(B^{\prime})=B^{\prime}_{P}$, for every $j<m$ we choose some
$L_{0}$-structure $B_{q_{j}}\in K_{0}$ containing $O(A)\cup b^{\prime}$ such
that $B_{q_{j}}\models\varphi^{\prime}_{j}(a^{\prime},b_{q_{j}}^{\prime})$,
for any other $q\in P(B^{\prime})$ we choose some $L_{0}$-structure $B_{q}\in
K_{0}$ containing $O(A)\cup b^{\prime}$ and then we build $O(B^{\prime})$ by
amalgamating all of these $L_{0}$-structure over $A\cup b^{\prime}$, we get
that $B^{\prime}\models\varphi(a^{\prime},b^{\prime})$.∎

######  Lemma 5.

The class $K$ satisfies the conditions of (Free Amalgamation) and (Generic
extension).

###### Proof.

Consider the structures $E\subseteq A,B\in K$. We will construct an
$L$-structure in $K$ satisfying the (Free Amalgamation) universal property.
Let us write $E=(O(E),P(E))$, $A=(O_{A},P_{A})$ and $B=(O_{B},P_{B})$. We
begin by forming the free amalgam $P(D):=P(A)\oplus_{P(E)}P(B)\in K_{P}$ which
will be the parameter sort of the free amalgam in $K$.

Let $(p_{i})_{i<\omega}$ enumerate $P(D)$. We define a sequence
$((D^{k}_{i})_{i<\omega})_{k<\omega}$ of tuples of countable
$L_{0}$-structures indexed by $P_{D}$ by induction. Let $D^{0}_{-1}:=A\cup B$
and let $E_{0}$ be the structure generated by the constants inside of $K_{0}$
(which is unique).

We define
$D^{0}_{p_{i}}:=(A_{p_{i}}\oplus_{E_{p_{i}}}B_{p_{i}})\oplus_{\emptyset}\langle
D^{0}_{p_{i-1}}\setminus(A\cup B)\rangle$ if $p_{i}\in P(E)$,
$D^{0}_{p_{i}}:=A_{p_{i}}\oplus_{\emptyset}\langle D^{0}_{p_{i-1}}\setminus
A\rangle$ if $p_{i}\in P(A)\setminus P(E)$,
$D^{0}_{p_{i}}:=B_{p_{i}}\oplus_{\emptyset}\langle D^{0}_{p_{i-1}}\setminus
B\rangle$ if $p_{i}\in P(B)\setminus P(E)$, and finally
$D^{0}_{p_{i}}:=E_{0}\oplus_{\emptyset}\langle D^{0}_{p_{i-1}}\rangle$ if
$p_{i}\in P(D)\setminus(P(A)\cup P(B))$. Then set
$D^{0}=\bigcup\limits_{i<\omega}D^{0}_{p_{i}}$. If $D^{k}$ is defined for some
$1\leq k<\omega$ we set
$D^{k+1}_{p_{0}}=D^{k}_{p_{0}}\oplus_{\emptyset}\langle D^{k}\setminus
D^{k}_{p_{0}}\rangle$, similarly if $D^{k}_{p_{i}}$ is defined for some $1\leq
i<\omega$ we set $D^{k+1}_{p_{i+1}}=D^{k}_{p_{i+1}}\oplus_{\emptyset}\langle
D^{k+1}_{i}\setminus D^{k}_{p_{i}}\rangle$.

Every $D^{k}_{p}$ is an $L_{0}$-structure for the parameter $p$. We set
$O(D)=\bigcup\limits_{k<\omega}D^{k}$, this set is countable and has an
$L_{0}$-structure for the parameter $p$ given by
$D_{p}=\bigcup\limits_{k<\omega}D^{k}_{p}\in K_{0}$, so $D\in K$. This defines
an $L$ structure $D=(O(D),P(D))\in K$ which extends $A$ and $B$, and such that
$A\cap B=E$ inside of $D$. Also if $A$ and $B$ are generated by
$(\overline{a},P(A))$ and $(\overline{b},P(B))$ respectively then
$(\overline{ab},P(A),P(B))$ generates $D$.

Now that we constructed the structure $D$ we show that is satisfies the right
property. Consider $D^{\prime}\in K$, $\varphi_{A}:A\rightarrow D^{\prime}$
and $\varphi_{B}:B\rightarrow D^{\prime}$ as in the statement, and write
$\varphi_{E}$ their restriction to $E$. The uniqueness of $\varphi$ is clear
from the fact that $D$ is generated by the images of $A$ and $B$. By free
amalgamation in $K_{P}$ there is a unique morphism $\varphi_{P}$ of
$K_{P}$-structures from $P(D)$ to $P(D^{\prime})$ such that the diagram
commutes. For existence we construct an increasing sequence of tuples of
morphism of $L_{0}$-structures indexed on $P(D)$ :
$((\varphi^{k}_{p_{i}}:D^{k}_{p_{i}}\rightarrow
D^{\prime}_{\varphi_{P}(p_{i})})_{i<\omega})_{k<\omega}$.

For $k=0$, if $p_{i}\in P(E)$ we take the unique morphism
$\varphi^{0}_{p_{i}}:D^{0}_{p_{i}}\rightarrow D^{\prime}$ that extends
$\varphi_{A}(p,\mathunderscore):A_{p}\rightarrow
D^{\prime}_{\varphi_{P}(p_{i})}$ and
$\varphi_{B}(p,\mathunderscore):B_{p}\rightarrow
D^{\prime}_{\varphi_{P}(p_{i})}$ and sends any element $d\in D^{0}_{p_{i-1}}$
to $\varphi^{0}_{p_{i-1}}(d)$. If $p_{i}\in P(A)\setminus P(E)$ we take the
unique morphism $\varphi^{0}_{p_{i}}:D^{0}_{p_{i}}\rightarrow
D^{\prime}_{\varphi_{P}(p_{i})}$ that extends
$\varphi_{A}(p,\mathunderscore):A_{p}\rightarrow
D^{\prime}_{\varphi_{P}(p_{i})}$ and sends any element $d\in D^{0}_{p_{i-1}}$
to $\varphi^{0}_{p_{i-1}}(d)$, similarly if $p_{i}\in P(B)\setminus P(E)$ we
take the unique morphism $\varphi^{0}_{p_{i}}:D^{0}_{p_{i}}\rightarrow
D^{\prime}_{\varphi_{P}(p_{i})}$ that extends
$\varphi_{B}(p,\mathunderscore):B_{p_{i}}\rightarrow
D^{\prime}_{\varphi_{P}(p_{i})}$ and sends any element $d\in D^{0}_{p_{i-1}}$
to $\varphi^{0}_{p_{i-1}}(d)$, and finally if $p_{i}\in P(D)\setminus(P(A)\cup
P(B))$ we take the unique morphism
$\varphi^{0}_{p_{i}}:D^{0}_{p_{i}}\rightarrow D^{\prime}_{\varphi_{P}(p_{i})}$
that sends any element $d\in D^{0}_{p_{i-1}}$ to $\varphi^{0}_{p_{i-1}}(d)$.

Assume that for every $i<\omega$ the functions
$\varphi^{k}_{p}:D^{k}_{p}\rightarrow D^{\prime}$ have been defined. Define
$\varphi^{k+1}_{p_{0}}:D^{k+1}_{p_{0}}\rightarrow D^{\prime}$ as the unique
morphism extending $\varphi^{k}_{p_{0}}:D^{k}_{p}\rightarrow D^{\prime}$ by
sending $x\in D^{k}_{p_{i}}$ to $\varphi^{k}_{p_{i}}(x)$ for every $i<\omega$.
We just repeat the same steps for the other parameters. Once all these
functions are constructed we consider
$\varphi=\bigcup\limits_{k<\omega}\varphi^{k}_{p}$ for every $p\in P(D)$. This
function defines a map in $K$ from $D$ to $D^{\prime}$ such that its
restrictions to $A$ and $B$ are $\varphi_{A}$ and $\varphi_{B}$ respectively,
and it is a morphism of $L_{0}$-structure since
$\varphi=\bigcup\limits_{k<\omega}\varphi^{k}_{p}:D_{p}\rightarrow
D^{\prime}_{\varphi(p)}$ is a morphism of $L_{0}$-structure for every
parameter $p\in P(D)$ and $\varphi_{P}:P(D)\rightarrow P(D^{\prime})$ is a
morphism of $L_{P}$-structure.

The proof of (Generic extension) carries over similarly.∎

######  Proposition 6.

$K$ satisfies ($3$ Amalgamation for algebraically independent extension).

###### Proof.

The proof carries over verbatim, we form the amalgam for the parameter sort to
find $P(D)$, then for every $p\in P(D)$ we choose some $L_{0}$-structures
$D^{0}_{p}$, using ($3$ Amalgamation for algebraically independent extension)
in $K_{0}$ when the parameter $p$ is in the basis, and we put all of these in
common by amalgamating them freely as we did in Lemma 5. ∎

###  2.3 Examples and non-examples

Some examples of classes $K_{0}$ satisfying $(H)$ are the class of countable
vector spaces over a finite (or countable) field (which is in the language),
the class of countable abelian groups, the class of countable sets with no
structure, and an other one is the class of countable graphs, in all those
cases the conditions 1-5 are easy to check, and the condition $6$ is a
consequence of the following proposition. In the case where the limit $T_{0}$
of the class $K_{0}$ is a stable theory we get that $\mathrel{\mathop{\vbox{
\hbox{\oalign{\kern-1.29167pt\cr\hfil$|$\hfil\cr\kern-3.01385pt\cr$\smile$\cr\kern-1.29167pt\cr}}
}}\displaylimits}^{a}=\mathrel{\mathop{\vbox{
\hbox{\oalign{\kern-1.29167pt\cr\hfil$|$\hfil\cr\kern-3.01385pt\cr$\smile$\cr\kern-1.29167pt\cr}}
}}\displaylimits}^{\Gamma}$.

######  Lemma 1.

If some class $K$ satisfies $1-5$ and that free independence and algebraic
independence coincide, i.e. that for every $E\subseteq A,B\in K$,
$A,B\subseteq D\in K$, the canonical map $\varphi:A\oplus_{E}B\rightarrow D$
is an embedding if and only if $A\cap B=E$ inside of $D$ (all the previous
examples satisfies this), then the condition $6$ holds and the limit $T$ is
stable.

###### Proof.

The hypothesis we have implies in particular that algebraic independence
satisfies base monotonicity in the structures of $K$ (by the same proof as the
fact that $\mathrel{\mathop{\vbox{
\hbox{\oalign{\kern-1.29167pt\cr\hfil$|$\hfil\cr\kern-3.01385pt\cr$\smile$\cr\kern-1.29167pt\cr}}
}}\displaylimits}^{\Gamma}$ satisfies base monotonicity in Proposition 10).
Given $E,A,B_{0},B_{1},D_{0}D_{1}$ and a diagram satisfying the assumptions of
$6$ consider the following diagram, where
$D=[D_{0}\oplus_{B_{0}}B]\oplus_{A\oplus_{E}B}[D_{1}\oplus_{B_{1}}B]$. We want
to show that $D_{0}\cap D_{1}=A$, $D_{0}\cap B=B_{0}$ and $D_{1}\cap B=B_{1}$
inside of $D$. We abuse notation and write $\langle AB\rangle$ to mean the
substructure of $D$ generated by the images of $A$ and $B$.

${D_{1}\oplus_{B_{1}}B}$${D}$${D_{1}}$${A\oplus_{E}B}$${D_{0}\oplus_{B_{0}}B}$${A}$${D_{0}}$${B_{1}}$${B}$${E}$${B_{0}}$

The maps form $A\oplus_{E}B$ to $D_{i}\oplus_{B_{1}}B$ come from the fact that
$A$ and $B$ are free over $E$ in $D_{i}\oplus_{B_{1}}B$ : in fact if some
element of $D_{i}\oplus_{B_{1}}B$ is in the image of both $A$ and $B$, then it
is also in the image of $D_{i}$, so it is in the image of $B_{i}$, and by
assumption $B_{i}$ and $A$ are free over $E$ in $D_{i}$. This diagram is
commutative and all arrows are embeddings. If some element of $D$ is in the
image of $D_{i}$ and $B$ then it is in the image of $B_{i}$. If some element
of $D$ is in the image of $D_{0}$ and $D_{1}$ then it is in the image of
$A\oplus_{E}B$ which is just the substructure generated by the images of $A$
and $B$. By $D_{i}\cap B=B_{i}$ and base monotonicity we have that
$D_{i}\cap(A\oplus_{E}B)=A\oplus_{E}B_{i}$ in $D$, so it is enough to show
that $\langle AB_{0}\rangle\cap\langle AB_{1}\rangle=A$. Now $A\cap B=E$ in
$D$ since $D_{0}\cap B=B_{0}$ and $A\cap B_{0}=E$, and $B_{0}\cap B_{1}=E$
implies that $\langle AB_{0}\rangle\cap\langle AB_{1}\rangle=A$, so we get
that $6$ holds.∎

We will describe now how adding generic structure to the class $K_{0}$ might
affect the conditions $(H)$ and some properties of the theory.

####  2.3.1 Adding a generic predicate

######  Definition 1.

Consider a new symbol $U$ for a unary predicate in some sort $S$, we define
$L_{U}=L\cup\\{U\\}$ and the class $K_{U}$ of $L_{U}$-structures consisting of
a structure $A$ in $K_{0}$ equipped with a subset $U(A)\subseteq S(A)$.

###### Proof.

The conditions $1-4$ are satisfied in $K_{U}$, we just take
$U(A\oplus_{E}B):=U(A)\cup U(B)$ and $U(A\oplus_{\emptyset}\langle
x\rangle):=U(A)$, similarly for $6$. For $5$ consider a finitely generated
structure $\langle a,b\rangle\in K_{U}$ and a quantifier free $L_{U}$ formula
$\varphi$ such that $\models\varphi(a,b)$. W.l.o.g. w.c.a. that
$\varphi(x,y)=\varphi^{\prime}(x,y)\wedge\bigwedge\limits_{i<n}U(t_{i}(x,y))\wedge\bigwedge\limits_{i<n}\neg
U(t^{\prime}_{i}(x,y))$ where $\varphi^{\prime}(x,y)$ is an $L_{0}$-formula
and the $t_{i}$ and $t^{\prime}_{i}$ are $L_{0}$-terms. By taking a
strengthening of $\varphi$ we can assume that
$\varphi^{\prime}(x,y)\vdash\bigwedge\limits_{j<n}t^{\prime}_{i}(x,y)\centernot=t_{j}(x,y)$
for every $i<n$. Now we consider the formula $\psi^{\prime}(x)$ associated to
the $L_{0}$-formula
$\varphi^{\prime}(x,y)\wedge\bigwedge\limits_{i,j<n}(t^{\prime}_{i}(x,y)\centernot=t_{j}(x,y))$
by $5$ in $K_{0}$. If for some $a^{\prime}\in A^{\prime}\in K_{R}$ we have
that $A^{\prime}\models\psi(a^{\prime})$ then by assumption we can find some
$b^{\prime}\in B^{\prime}$ an extension of $A^{\prime}$ in $K_{0}$ such that
$B^{\prime}\models\varphi^{\prime}(a^{\prime},b^{\prime})\wedge\bigwedge\limits_{i<n}(t^{\prime}_{i}(a^{\prime},b^{\prime})\centernot\in\langle
t_{j}(a^{\prime},b^{\prime}))_{j<n}\rangle)$. Define the subset $U$ on
$B^{\prime}$ to be
$U(A^{\prime})\cup\\{t_{j}(a^{\prime},b^{\prime}))_{j<n}\\}$, we get that
$B^{\prime}\models\neg U(t^{\prime}_{i}(a^{\prime},b^{\prime}))$ for all
$i<n$, so $B^{\prime}\models\varphi(a^{\prime},b^{\prime})$, and $B^{\prime}$
is an extension of $A^{\prime}$ in $K_{U}$.∎

######  Remark 2.

If there is some $a,b\in A\in K$ and $c=t(a,b)\in\langle
ab\rangle\setminus(\langle a\rangle\cup\langle b\rangle)$ the theory $T_{U}$
of a limit of $K_{U}$ is non stable, in fact the formula $U(t(x,y))$ would
then have the order property.

######  Definition 3.

Let $L_{R}=L\cup\\{R\\}$ a new unary predicate symbol. We define the class
$K_{R}$ of $L_{R}$-structures consisting of a structure $A$ in $K_{0}$
equipped with a substructure $R(A)$.

######  Proposition 4.

If the structures of $K_{0}$ are uniformly locally finite the class $K_{R}$
satisfies $1-5$. However if the generated structure in $K_{0}$ is non trivial
the condition $(6)$ need not to hold.

###### Proof.

The conditions $1-4$ are still true by the same arguments as before, we will
need some assumption on $K_{0}$ to get $5$. We take $P(A\oplus_{E}B):=\langle
P(A)P(B)\rangle$ and $P(A\oplus_{\emptyset}\langle x\rangle):=P(A)$. For $5$
consider a finitely generated structure $\langle a,b\rangle\in K_{R}$ and a
quantifier free $L_{R}$ formula $\varphi$ such that $\langle
a,b\rangle\models\varphi(a,b)$. W.l.o.g. w.c.a. that
$\varphi(x,y)=\varphi^{\prime}(x,y)\wedge\bigwedge\limits_{i<n}R(t_{i}(x,y))\wedge\bigwedge\limits_{i<n}\neg
R(t^{\prime}_{i}(x,y))$ where $\varphi^{\prime}(x,y)$ is an $L_{0}$-formula,
for all $c\in R(\langle ab\rangle)$ there is some $i<n$ such that
$c=t_{i}(a,b)$ and for all $c\in\langle ab\rangle\setminus R(\langle
ab\rangle)$ there is some $i<n$ such that $c=t^{\prime}_{i}(a,b)$. Since our
theory is uniformly locally finite the condition
$t^{\prime}_{i}(x,y)\centernot\in\langle(t_{j}(x,y))_{j<n}\rangle$ is
definable. Now we consider the formula $\psi^{\prime}(x)$ associated to the
$L_{0}$-formula
$\varphi^{\prime}(x,y)\wedge\bigwedge\limits_{i<n}(t^{\prime}_{i}(x,y)\centernot\in\langle(t_{j})_{j<n}\rangle)$
by $5$ in $K_{0}$, and then the formula
$\psi(x):=\psi^{\prime}(x)\wedge\bigwedge\limits_{k<m}R(t_{i_{k}}(x))\wedge\bigwedge\limits_{k<m}\neg
R(t_{i_{k}}(x))$, where $t_{i_{k}}(x)$ enumerates the terms $t$ such that
$R(t(a))$ and $t^{\prime}_{i_{k}}(x)$ enumerates the terms $t^{\prime}$ such
that $\neg R(t^{\prime}(a))$. Now if for some $a^{\prime}\in A^{\prime}\in
K_{R}$ we have that $A^{\prime}\models\psi(a^{\prime})$ then by assumption we
can find some $b^{\prime}$ such that $\langle
a^{\prime}b^{\prime}\rangle\models\varphi^{\prime}(a^{\prime},b^{\prime})\wedge\bigwedge\limits_{i<n}(t^{\prime}_{i}(a^{\prime},b^{\prime})\centernot\in\langle
t_{j}(a^{\prime},b^{\prime}))_{j<n}\rangle)$. Define the substructure $R$ on
$\langle a^{\prime}b^{\prime}\rangle$ to be $\langle
t_{j}(a^{\prime},b^{\prime}))_{j<n}\rangle$, we get that $\langle
a^{\prime}b^{\prime}\rangle\models\neg
R(t^{\prime}_{i}(a^{\prime},b^{\prime}))$ for all $i<n$, so $\langle
a^{\prime}b^{\prime}\rangle\models\varphi(a^{\prime},b^{\prime})$, and this
property still holds in $B^{\prime}:=A^{\prime}\oplus_{\langle
a^{\prime}\rangle}\langle a^{\prime}b^{\prime}\rangle$, which is an extension
of $A$ as an $L_{R}$-structure.

Now about $6$ : By $6$ in the class $K_{0}$ we can find some structure $D\in
K_{0}$ and some embeddings as in the statement. We want to define $R(D)$. We
know that in any case $\langle R(D_{0})\cup R(D_{1})\cup R(B)\langle\subseteq
R(D)$, however it might happen that for some $b\in(B\setminus R(B))$ we have
that $b\in\langle R(D_{0})\cup R(D_{1})\cup R(B)\rangle$, and in that case the
$L_{0}$-embedding $\varphi_{B}$ can not be an $L_{R}$-embedding.∎

To give a simple example of this last fact, let us look at some vector space
over a finite field with some a predicate for a subspace. Take some
independent vectors $a_{0},a_{1},b_{0},b_{1}$ as in the statement of the
independence theorem for $\mathrel{\mathop{\vbox{
\hbox{\oalign{\kern-1.29167pt\cr\hfil$|$\hfil\cr\kern-3.01385pt\cr$\smile$\cr\kern-1.29167pt\cr}}
}}\displaylimits}^{a}$ such that $\neg R(a_{0}+b_{0})$, $R(a_{1}-b_{1})$ and
$R(b_{0}+b_{1})$ are satisfied. Then if $a$ satisfies $\mbox{\rm
tp}(a_{i}/b_{i})$ for $i=0,1$ we have that $R(a+b_{0})$ is satisfied, which is
a contradiction. In that case a limit of $K_{R}$ still exist, and it might be
NSOP1, however in that case we have that $\mathrel{\mathop{\vbox{
\hbox{\oalign{\kern-1.29167pt\cr\hfil$|$\hfil\cr\kern-3.01385pt\cr$\smile$\cr\kern-1.29167pt\cr}}
}}\displaylimits}^{K}\centernot=\mathrel{\mathop{\vbox{
\hbox{\oalign{\kern-1.29167pt\cr\hfil$|$\hfil\cr\kern-3.01385pt\cr$\smile$\cr\kern-1.29167pt\cr}}
}}\displaylimits}^{a}$ in this limit.

######  Remark 5.

If we assume that the class $K$ is uniformly locally finite we can restrict it
to the class of finite structures in $K$ and have a Fraïssé class in the usual
sense, to each isomorphism type of the structure generated by the constants
correspond a limit, and this limit is $\omega$-categorical.

As we have seen in the previous section the condition $6$ puts a strong
restriction in what we can add to the structure, and some classes with an
NSOP1 limit can satisfy the conditions $1$ to $5$ and not satisfy $6$. For
example the class of finite equivalence relation, with limit the (stable)
theory of the equivalence relation with an infinite number of classes all
infinite (which we will write $T_{\infty}$), or its parameterized version (see
[15] or [3]) which is strictly NSOP1, satisfies existence and that
$\mathrel{\mathop{\vbox{
\hbox{\oalign{\kern-1.29167pt\cr\hfil$|$\hfil\cr\kern-3.01385pt\cr$\smile$\cr\kern-1.29167pt\cr}}
}}\displaylimits}^{K^{M}}=\mathrel{\mathop{\vbox{
\hbox{\oalign{\kern-1.29167pt\cr\hfil$|$\hfil\cr\kern-3.01385pt\cr$\smile$\cr\kern-1.29167pt\cr}}
}}\displaylimits}^{f}=\mathrel{\mathop{\vbox{
\hbox{\oalign{\kern-1.29167pt\cr\hfil$|$\hfil\cr\kern-3.01385pt\cr$\smile$\cr\kern-1.29167pt\cr}}
}}\displaylimits}^{d}$ over arbitrary sets.

######  Definition 6.

Consider a new symbol $E$ for a binary predicate in some sort $S$, we define
the language $L_{E}=L\cup\\{V,E,p\\}$ where $V$ is a new sort, $E$ a binary
relation on $S$, and $p$ a unary map form $S$ to $V$. Define the class $K_{E}$
of $L_{E}$-structures consisting of a structure $A$ in $K_{0}$ equipped with
an equivalence relation $E^{A}\subseteq S(A)^{2}$, a new sort $V(A)$ and a map
$p:S(A)\rightarrow V(A)$ realizing the quotient.

######  Proposition 7.

The class $K_{E}$ satisfies $(H)$.

###### Proof.

If we do not add the quotient to the structure the conditions $1-5$ are
satisfied but condition $6$ is not, also if we ask for the map $p$ to be
surjective the condition $3$ will not hold. It is easy to see that the
conditions $1-5$ are satisfied in $K_{E}$ : the only structure on the quotient
is equality and we can do ’in parallel’ free amalgamation in $K$ and in the
quotient. For $6$ we proceed similarly by forming the amalgam in $K$ on one
end and the amalgam of the quotient on the other end.∎

This construction is the same as adding a new sort $V$ with no structure other
than equality and then some function from $S$ to $V$.

####  2.3.2 Adding generic functions

######  Definition 1.

Consider a new symbol $f$ for a binary function, we define
$L_{f}=L\cup\\{f\\}$ and the class $K_{f}$ of $L_{f}$-structures consisting of
a structure $A$ in $K_{0}$ equipped with a binary function $f_{A}:A\times
A\rightarrow A$.

######  Proposition 2.

The class $K_{f}$ satisfies $(H)$.

###### Proof.

The conditions $1$ and $4$ are clearly satisfied in $K_{f}$. For the condition
$3$ we construct some $L_{f}$-structure $D$ using the properties of $K$.
Assume we have our two extensions of $L_{f}$-structures $E\subseteq A,B$. We
define an increasing sequence of $L_{0}$-structures $(D_{i})_{i<\omega}$
equipped with a partial binary function $f_{i+1}:D_{i}^{2}\rightarrow
D_{i+1}$. Let $D_{0}:=A\oplus_{E}B$ and $f_{0}=f_{E}$. Now set
$D_{1}:=D_{0}\oplus_{\emptyset}\langle D_{0}^{2}\setminus(A^{2}\cup
B^{2})\rangle$ and consider $f_{1}:D_{0}^{2}\rightarrow D_{1}$ defined the
following way : if $(x,y)\in A^{2}$, $f_{1}(x,y):=f_{A}(x,y)$, if $(x,y)\in
B^{2}$, $f_{1}(x,y):=f_{B}(x,y)$ and else $(x,y)$ is sent to the element of
$D_{1}$ with the corresponding label. We iterate this process : if
$D_{i},f_{i+1}$ are defined for $i>0$ we let
$D_{i+1}:=D_{i}\oplus_{\emptyset}\langle
D_{i}^{2}\setminus(D_{i-1}^{2})\rangle$, and $f_{i+1}$ extends $f_{i}$ by
sending $(x,y)\in D_{i}^{2}\setminus(D_{i-1}^{2})$ to the element of $D_{i+1}$
with the corresponding label. We then set $D:=\bigcup\limits_{i<\omega}D_{i}$
and $f:=\bigcup\limits_{i<\omega}f_{i}:D^{2}\rightarrow D$. This structure
satisfies the universal property : it is generated by the images of $A$ and
$B$ as an $L_{f}$-structure, given $D^{\prime}$ and embeddings
$\varphi_{A}:A\rightarrow D^{\prime}$, $\varphi_{B}:B\rightarrow D^{\prime}$
as in the statement we can define an increasing sequence of morphism of
$L_{0}$-structures $\varphi_{i}:D_{i}\rightarrow D^{\prime}$ in a canonical
way and show that its union is a morphism of $L_{f}$-structure. The
construction of $A\oplus_{\emptyset}\langle x\rangle$ follows the same method.

The condition $6$ is easy to check, we just apply $6$ inside of $K_{0}$, find
the embeddings, take $f_{D}|_{D_{i}^{2}}$ to be the image of $f_{D_{i}}$ by
$\varphi_{D_{i}}$, similarly for $B$, and we can set $f$ to be anything
outside of $D^{2}_{i}$ and $B^{2}$. Now for condition $5$ consider a structure
$\langle ab\rangle\in K_{f}$ and some qf-$L_{f}$-formula $\varphi$ such that
$\langle ab\rangle\models\varphi(a,b)$. By replacing every instance of $f$ in
$\varphi$ by a new variable and separating the instances of $f$ applied only
to elements of $\langle a\rangle$ from the other ones we can reshape
$\varphi(x,y)$ in the following form :

$\varphi^{\prime}(x,x_{\leq m},y,y_{\leq n})\wedge\bigwedge\limits_{i\leq
m}(x_{i}=f(s_{i}(x,x_{<i}),s^{\prime}_{i}(x,x_{<i})))\wedge\bigwedge\limits_{i\leq
n}(y_{i}=f(t_{i}(x,x_{\leq m},y,y_{<i}),t^{\prime}_{i}(x,x_{\leq
m},y,y_{<i})))$

$\varphi^{\prime}$ is an $L_{0}$-qf-formula and the
$t_{i},s_{i},t^{\prime}_{i},s^{\prime}_{i}$ are $L_{0}$-terms. By
strengthening the formula $\varphi$ we can assume that, if we define by
induction $x_{k}(a)=f(s_{k}(a,x_{<k}(a)),s^{\prime}_{k}(a,x_{<k}(a)))$ and

$y_{k}(a,b):=f(t_{i}(a,x_{\leq m}(a),b,y_{<i}(a,b)),t^{\prime}_{i}(a,x_{\leq
m}(a),b,y_{<i}(a,b)))$

, and if $y_{k}(a,b)\centernot=y_{k^{\prime}}(a,b)$, then
$\varphi^{\prime}(x,x_{\leq m},y,y_{\leq n})$ specifies that $(t_{k}(x,x_{\leq
m},y,y_{<k}),t^{\prime}_{k}(x,x_{\leq m},y,y_{<k}))\centernot=(t_{k}(x,x_{\leq
m},y,y_{<k}),t^{\prime}_{k}(x,x_{\leq m},y,y_{<k}))$, and similarly for
$x_{k},x_{k}^{\prime}$ and $x_{k},y_{k^{\prime}}$.

Now consider the $L_{0}$-formula $\psi^{\prime}(x,x_{\leq m})$ associated by
$5$ in $K_{0}$ to $\varphi^{\prime}(x,x_{\leq m},y,y_{\leq n})$ and then the
$L_{f}$-formula $\psi^{\prime}(x):=\psi^{\prime}(x,x_{\leq
m})\wedge\bigwedge\limits_{i\leq
m}(x_{i}=f(s_{i}(x,x_{<i}),s^{\prime}_{i}(x,x_{<i})))$. Assume that for some
$a^{\prime}\in A^{\prime}$ we have that $A^{\prime}\models\psi(a^{\prime})$.
By assumption there is an $L_{0}$-extension $A^{\prime}\subseteq B^{\prime}$
and $b^{\prime},b^{\prime}_{\leq n}\in B^{\prime}$ such that
$B^{\prime}\models\varphi^{\prime}(a^{\prime},a^{\prime}_{\leq
m},b^{\prime},b^{\prime}_{\leq n})\wedge\bigwedge\limits_{i\leq
m}(a^{\prime}_{i}=f(s_{i}(a^{\prime},a^{\prime}_{<i}),s^{\prime}_{i}(a^{\prime},a^{\prime}_{<i})))$.
We just need to define a binary function $f_{B^{\prime}}$ on $B^{\prime}$
extending $f_{A^{\prime}}$ such that
$b^{\prime}_{k}=f(t_{i}(a^{\prime},a^{\prime}_{\leq
m},b^{\prime},b^{\prime}_{<i}),t^{\prime}_{i}(a^{\prime},a^{\prime}_{\leq
m},b^{\prime},b^{\prime}_{<i})$ for every $k\leq n$. We begin by setting these
values, this is possible thanks to the strengthening we had, and we can let
$f_{B^{\prime}}$ be whatever outside of this, we write $B^{\prime}_{f}$ for
the $L_{f}$-structure defined on $B^{\prime}$, then
$B^{\prime}_{f}\models\varphi(a^{\prime},b^{\prime})$.∎

######  Remark 3.

Adding a generic binary function assures us that the theory of any limit is
strictly NSOP1, as proven in [18, Section 3]. Also we get that the generated
structure is non-trivial. With the same construction we can add generic
functions of any arity.

######  Definition 4.

We define $L_{\pi}=L\cup\\{\pi,\pi^{-1}\\}$ new symbols of unary functions and
the class $K_{\pi}$ of $L_{\pi}$-structures consisting of a structure $A$ in
$K_{0}$ equipped with a bijection $\pi_{A}$ of inverse function
$\pi^{-1}_{A}$.

######  Proposition 5.

The class $K_{\pi}$ satisfies $(H)$.

###### Proof.

The conditions $1$ and $4$ are clearly satisfied in $K_{\pi}$, for the
conditions $5$ and $6$ we can proceed as previously. Now we describe what to
pick as the free amalgam for $2$. Let $E\subseteq A,B$ be two extensions in
$K_{\pi}$.

In the same fashion as previously we construct an increasing sequence of
$L_{0}$-structures $(D_{i})_{i<\omega}$, and an increasing sequence of subsets
$H_{i}\subseteq D_{i}$ and two increasing sequences of functions
$\pi_{i},\pi_{i}^{-1}:H_{i}\rightarrow D_{i}$ such that $D_{i}\subseteq
H_{i+1}$ and
$\pi_{i+1}\cdot\pi^{-1}_{i}=\pi_{i+1}^{-1}\cdot\pi_{i}=id_{H_{i}}$ for every
$i<\omega$. Let $D_{0}:=A\oplus_{E}B$, $H_{0}=A\cup B$, and
$\pi_{0}=\pi_{A}\cup\pi_{B}$, $\pi^{-1}_{0}=\pi^{-1}_{A}\cup\pi^{-1}_{B}$.
$D_{1}:=D_{0}\oplus_{\emptyset}\langle[D_{0}\setminus(A\cup
B)]_{k\in\mathbb{Z}\setminus\\{0\\}}\rangle$, i.e., we consider the
$L_{0}$-structure freely generated over $D_{0}$ by copies of
$D_{0}\setminus(A\cup B)$ indexed on $\mathbb{Z}\setminus\\{0\\}$, we see the
element of $D_{0}\setminus(A\cup B)$ as those corresponding to the copy of
index $0$. Set
$H_{1}=D_{0}\cup\bigcup\limits_{k\in\mathbb{Z}\setminus\\{0\\}}[D_{0}\setminus(A\cup
B)]_{k}$. We define $\pi_{1}$ as the extension of $\pi_{0}$ that send an
element of $[D_{0}\setminus(A\cup B)]_{k}$ to its copy of index $k+1$, i.e. it
sends $t(a,b)_{k}$ to $t(a,b)_{k+1}$, similarly for $\pi_{0}^{-1}$ which sends
an element of $[D_{0}\setminus(A\cup B)]_{k}$ to its copy of index $k-1$.
Assume that $D_{i+1}$,$H_{i}$ and $\pi_{i},\pi_{i}^{-1}$ have been defined for
$i>0$. We set $D_{i+1}:=D_{i}\oplus_{\emptyset}\langle[D_{i}\setminus
D_{i-1}]_{i\in\mathbb{Z}\setminus\\{0\\}}\rangle$,
$H_{i+1}:=D_{i}\cup\bigcup\limits_{k<\omega}[D_{i}\setminus D_{i-1}]_{k}$ and
take $\pi_{i+1}$ to be the function extending $\pi_{i}$ that sends an element
of $[D_{i}\setminus D_{i-1}]_{k}$ to its copy of index $k+1$, similarly for
$\pi_{0}^{-1}$ which sends an element of $[D_{i}\setminus D_{i-1}]_{k}$ to its
copy of index $k-1$. Once this is done we take
$D:=\bigcup\limits_{i<\omega}D_{i}$,
$\pi_{D}:=\bigcup\limits_{i<\omega}\pi_{i}$ and
$\pi^{-1}_{D}:=\bigcup\limits_{i<\omega}\pi^{-1}_{i}$. This defines an
$L_{\pi}$-structure of bijection $\pi_{D}$, and $D$ satisfies the universal
property : it is generated by the images of $A$ and $B$ as an
$L_{\pi}$-structure, given $D^{\prime}$ and embeddings
$\varphi_{A}:A\rightarrow D^{\prime}$, $\varphi_{B}:B\rightarrow D^{\prime}$
as in the statement we can define an increasing sequence of morphism of
$L_{0}$-structures $\varphi_{i}:D_{i}\rightarrow D^{\prime}$ in a canonical
way and show that its union is a morphism of $L_{\pi}$-structure. We take
$\varphi_{i}:D_{0}\rightarrow D^{\prime}$ to be the only morphism of
$L_{0}$-structure such that $\varphi_{i}|_{A}=\varphi_{A}$ and
$\varphi_{i}|_{B}=\varphi_{B}$. If $\varphi_{i}:D_{0}\rightarrow D^{\prime}$
is defined we take $\varphi_{i+1}:D_{i+1}\rightarrow D^{\prime}$ to be its
unique morphism extending $\varphi_{i}$ that sends $\pi_{D}^{k}(t)$ to
$\pi^{k}_{D^{\prime}}(\varphi(t))$ for every $t\in D_{i}$. Set
$\varphi:=\bigcup\limits_{i<\omega}\varphi_{i}$, it is a morphism of
$L_{0}$-structure and by construction it commutes with $\pi$, so it is a
morphism of $L_{\pi}$-structures. The construction of
$A\oplus_{\emptyset}\langle x\rangle$ follows the same method. ∎

######  Definition 6.

We define $L_{\sigma}=L\cup\\{\sigma,\sigma^{-1}\\}$ new symbols of unary
functions and the class $K_{\sigma}$ of $L_{\sigma}$-structures consisting of
a structure $A$ in $K_{0}$ equipped with an automorphism $\sigma_{A}$ of
inverse function $\sigma^{-1}_{A}$.

In $K_{\sigma}$ the condition $1-4$ are easy to check : we use the universal
property to show that if $E\subseteq A,B\in K_{\sigma}$ then there is an
(unique) automorphism $\sigma_{A\oplus_{E}B}$ of $A\oplus_{E}B$ extending
$\sigma_{A}$ and $\sigma_{B}$ and $(A\oplus_{E}B,\sigma_{A\oplus_{E}B})$
satisfies the universal property. For $3$ we consider the $L_{0}$-structure
$A\oplus_{\emptyset}\langle(x_{i})_{i\in\mathbb{Z}}\rangle$ equipped with the
unique automorphism that extends $\sigma_{A}$ and sends $x_{k}$ to $x_{k+1}$
for every $k\in\mathbb{Z}$. As for the generic substructure we need the
structures of $K_{0}$ to be uniformly locally finite for $5$ to hold, in this
case the proof follows the same method. Also the condition $6$ need not to
hold even in this case for similar reasons as in $K_{R}$. A limit of
$K_{\sigma}$ still exist, and it might be NSOP1, however in that case we have
that $\mathrel{\mathop{\vbox{
\hbox{\oalign{\kern-1.29167pt\cr\hfil$|$\hfil\cr\kern-3.01385pt\cr$\smile$\cr\kern-1.29167pt\cr}}
}}\displaylimits}^{K}\centernot=\mathrel{\mathop{\vbox{
\hbox{\oalign{\kern-1.29167pt\cr\hfil$|$\hfil\cr\kern-3.01385pt\cr$\smile$\cr\kern-1.29167pt\cr}}
}}\displaylimits}^{a}$ in this limit.

####  2.3.3 Some questions

  1. 1.

Can we give a more general context that could fit in the case of classes
satisfying $(H)$ and the case of Fraïssé classes satisfying S.A.P. in a
relational language from [10, Section 6.3], allowing $\mathrel{\mathop{\vbox{
\hbox{\oalign{\kern-1.29167pt\cr\hfil$|$\hfil\cr\kern-3.01385pt\cr$\smile$\cr\kern-1.29167pt\cr}}
}}\displaylimits}^{K}$ to be different from $\mathrel{\mathop{\vbox{
\hbox{\oalign{\kern-1.29167pt\cr\hfil$|$\hfil\cr\kern-3.01385pt\cr$\smile$\cr\kern-1.29167pt\cr}}
}}\displaylimits}^{a}$ in $T^{eq}$, by relaxing $6$ for example.

  2. 2.

Let $K$ be a class satisfying $(H)$ and $T$ be the theory of its limit. Is a
group definable in the limit $T_{P}$ of the parameterization of $K$ (without
any structure on the parameters) necessarily definably isomorphic to a product
of groups definable in $T$?

####  2.3.4 Some strictly NSOP1 theories with strong amalgamation

The theory $T$ of a limit of some $K$ satisfying $(H)$ is not a free
amalgamation theory in the sense of [11], our strong independence relation
$\mathrel{\mathop{\vbox{
\hbox{\oalign{\kern-1.29167pt\cr\hfil$|$\hfil\cr\kern-3.01385pt\cr$\smile$\cr\kern-1.29167pt\cr}}
}}\displaylimits}^{\Gamma}$ does not in general satisfies the closure
condition (which is defined as : if $C\subseteq A,B$ are algebraically closed
and $A\mathrel{\mathop{\vbox{
\hbox{\oalign{\kern-1.29167pt\cr\hfil$|$\hfil\cr\kern-3.01385pt\cr$\smile$\cr\kern-1.29167pt\cr}}
}}\displaylimits}_{C}B$ then $A\cup B$ is algebraically closed). The proof of
Corollary 11 can be generalised to show that in any theory $T$ with some
stationary independence relation $\mathrel{\mathop{\vbox{
\hbox{\oalign{\kern-1.29167pt\cr\hfil$|$\hfil\cr\kern-3.01385pt\cr$\smile$\cr\kern-1.29167pt\cr}}
}}\displaylimits}$ satisfying Proposition 10 and such that
$\mathrel{\mathop{\vbox{
\hbox{\oalign{\kern-1.29167pt\cr\hfil$|$\hfil\cr\kern-3.01385pt\cr$\smile$\cr\kern-1.29167pt\cr}}
}}\displaylimits}^{a}$ satisfies the independence theorem over arbitrary sets,
$\mathrel{\mathop{\vbox{
\hbox{\oalign{\kern-1.29167pt\cr\hfil$|$\hfil\cr\kern-3.01385pt\cr$\smile$\cr\kern-1.29167pt\cr}}
}}\displaylimits}$ implies $\mathrel{\mathop{\vbox{
\hbox{\oalign{\kern-1.29167pt\cr\hfil$|$\hfil\cr\kern-3.01385pt\cr$\smile$\cr\kern-1.29167pt\cr}}
}}\displaylimits}^{f}$, so that $T$ is $NSOP1$, has existence and
$\mathrel{\mathop{\vbox{
\hbox{\oalign{\kern-1.29167pt\cr\hfil$|$\hfil\cr\kern-3.01385pt\cr$\smile$\cr\kern-1.29167pt\cr}}
}}\displaylimits}^{a}=\mathrel{\mathop{\vbox{
\hbox{\oalign{\kern-1.29167pt\cr\hfil$|$\hfil\cr\kern-3.01385pt\cr$\smile$\cr\kern-1.29167pt\cr}}
}}\displaylimits}^{K}$ over arbitrary sets.