From 14de439d475ec315ae40c85f813036e5568a019f Mon Sep 17 00:00:00 2001
From: Franciszek Malinka <franciszek.malinka@gmail.com>
Date: Wed, 29 Jun 2022 16:03:50 +0200
Subject: =?UTF-8?q?Zmiana=20struktury,=20teraz=20osobne=20pliki=20na=20ka?=
 =?UTF-8?q?=C5=BCd=C4=85=20sekcj=C4=99?=
MIME-Version: 1.0
Content-Type: text/plain; charset=UTF-8
Content-Transfer-Encoding: 8bit

---
 sections/fraisse_classes.tex | 451 +++++++++++++++++++++++++++++++++++++++++++
 1 file changed, 451 insertions(+)
 create mode 100644 sections/fraisse_classes.tex

(limited to 'sections/fraisse_classes.tex')

diff --git a/sections/fraisse_classes.tex b/sections/fraisse_classes.tex
new file mode 100644
index 0000000..ca2247e
--- /dev/null
+++ b/sections/fraisse_classes.tex
@@ -0,0 +1,451 @@
+\documentclass[../lic_malinka.tex]{subfiles}
+
+\begin{document}
+  In this section we will take a closer look at classes of finitely
+  generated structures with some characteristic properties. More
+  specifically, we will describe a concept developed by a French
+  mathematician Roland Fraïssé called Fraïssé limit.
+
+  \subsection{Definitions} 
+  \begin{definition}
+    Let $L$ be a signature and $M$ be an $L$-structure. The \emph{age} of $M$ is
+    the class $\bK$ of all finitely generated structures that embeds into $M$.
+    The age of $M$ is also associated with class of all structures embeddable in
+    $M$ \emph{up to isomorphism}.
+  \end{definition}
+
+  \begin{definition}
+    We say that $M$ has \emph{countable age} when its age has countably many 
+    isomorphism types of finitely generated structures. 
+  \end{definition}
+
+  \begin{definition}
+    Let $\bK$ be a class of finitely generated structures. $\bK$ has 
+    \emph{hereditary property (HP)} if for any $A\in\bK$, any finitely generated
+    substructure $B$ of $A$ it holds that $B\in\bK$. 
+  \end{definition}
+
+  \begin{definition}
+    Let $\bK$ be a class of finitely generated structures. We say that $\bK$ has
+    \emph{joint embedding property (JEP)} if for any $A, B\in\bK$ there is a
+    structure $C\in\bK$ such that both $A$ and $B$ embed in $C$.
+
+    \begin{center}
+      \begin{tikzcd}
+                                  & C & \\
+        A \arrow[ur, dashed, "f"] &   & B \arrow[ul, dashed, "g"'] 
+	  \end{tikzcd}
+    \end{center}
+
+	In terms of category theory, this is a \emph{span} in category $\bK$.
+  \end{definition}
+
+  Fraïssé has shown fundamental theories regarding age of a structure, one of 
+  them being the following one:
+  
+  \begin{fact}
+    \label{fact:age_is_hpjep}
+    Suppose $L$ is a signature and $\bK$ is a nonempty finite or countable set 
+    of finitely generated $L$-structures. Then $\bK$ has the HP and JEP if
+    and only if $\bK$ is the age of some finite or countable structure. 
+  \end{fact}
+
+  Beside the HP and JEP Fraïssé has distinguished one more property of the 
+  class $\bK$, namely amalgamation property.
+
+  \begin{definition}
+    Let $\bK$ be a class of finitely generated $L$-structures. We say that $\bK$
+    has the \emph{amalgamation property (AP)} if for any $A, B, C\in\bK$ and
+    embeddings $e\colon C\to A, f\colon C\to B$ there exists $D\in\bK$ together
+    with embeddings $g\colon A\to D$ and $h\colon A\to D$ such that 
+    $g\circ e = h\circ f$.
+    \begin{center}
+      \begin{tikzcd}
+                                  & D & \\
+        A \arrow[ur, dashed, "g"] &   & B \arrow[ul, dashed, "h"'] \\
+                                  & C \arrow[ur, "f"'] \arrow[ul, "e"]
+      \end{tikzcd}
+    \end{center}
+  \end{definition}
+
+  In terms of category theory, amalgamation over some structure $C$ is a
+  pushout diagram.
+
+  \begin{definition}
+    Let $M$ be an $L$-structure. $M$ is \emph{ultrahomogeneous} if every
+    isomorphism between finitely generated substructures of $M$ extends to an
+    automorphism of $M$.
+  \end{definition}
+
+  Having those definitions we can provide the main Fraïssé theorem.
+
+  \begin{theorem}[Fraïssé theorem]
+    \label{theorem:fraisse_thm}
+    Let L be a countable language and let $\bK$ be a nonempty countable set of
+    finitely generated $L$-structures which has HP, JEP and AP. Then $\bK$ is 
+    the age of a countable, ultrahomogeneous $L$-structure $M$. Moreover, $M$ is
+    unique up to isomorphism. We say that $M$ is a \emph{Fraïssé limit} of $\bK$
+    and denote this by $M = \Flim(\bK)$.
+  \end{theorem}
+
+  This is a well known theorem. One can read a proof of this theorem in Wilfrid
+  Hodges' classical book \textit{Model Theory}~\cite{hodges_1993}. In the proof
+  of this theorem appears another, equally important \ref{lemma:weak_ultrahom}.
+
+  \begin{definition}
+    We say that an $L$-structure $M$ is \emph{weakly ultrahomogeneous} if for any
+    $A, B$ finitely generated substructures of $M$ such that $A\subseteq B$ and
+    an embedding $f\colon A\to M$ there is an embedding $g\colon B\to M$ which
+    extends $f$.
+    \begin{center}
+      \begin{tikzcd}
+        A \arrow[d, "\subseteq"'] \arrow[r, "f"] & D \\
+        B \arrow[ur, dashed, "g"']
+      \end{tikzcd}
+    \end{center}
+  \end{definition}
+
+  \begin{lemma}
+    \label{lemma:weak_ultrahom}
+    A countable structure is ultrahomogeneous if and only if it is weakly
+    ultrahomogeneous.
+  \end{lemma}
+
+  This lemma will play a major role in the later parts of the paper. Weak
+  ultrahomogeneity is an easier and more intuitive property and it will prove
+  useful when recursively constructing the generic automorphism of a Fraïssé 
+  limit.
+
+  % \begin{fact} If $\bK$ is a uniformly locally finite Fraïssé class, then
+  % $\Flim(\bK)$ is $\aleph_0$-categorical and has quantifier elimination.
+  % \end{fact}
+
+  \subsection{Random graph} 
+  
+  In this section we'll take a closer look on a class of finite unordered graphs, 
+  which is a Fraïssé class. 
+
+  The language of unordered graphs $L$ consists of a single binary
+  relational symbol $E$. If $G$ is an $L$-structure then we call it a
+  \emph{graph}, and its elements $\emph{vertices}$. If for some vertices 
+  $u, v\in G$ we have $G\models uEv$ then we say that there is an $\emph{edge}$ 
+  connecting $u$ and $v$. If $G\models \forall x\forall y (xEy\leftrightarrow yEx)$
+  then we say that $G$ is an \emph{unordered graph}. From now on we omit the word
+  \textit{unordered} and  graphs as unordered.
+
+  \begin{proposition}
+    Let $\cG$ be the class of all finite graphs. $\cG$ is a Fraïssé class.
+  \end{proposition}
+  \begin{proof}
+    $\cG$ is of course countable (up to isomorphism) and has the HP 
+    (graph substructure is also a graph). It has JEP: having two finite graphs 
+    $G_1,G_2$ take their disjoint union $G_1\sqcup G_2$ as the extension of them
+    both. $\cG$ has the AP. Having graphs $A, B, C$, where $B$ and $C$ are
+    supergraphs of $A$, we can assume without loss of generality, that 
+    $(B\setminus A) \cap (C\setminus A) = \emptyset$. Then 
+    $A\sqcup (B\setminus A)\sqcup (C\setminus A)$ is the graph we're looking
+    for (with edges as in B and C and without any edges between $B\setminus A$
+	and $C\setminus A$).
+  \end{proof}
+
+  \begin{definition}
+    The \emph{random graph} is the Fraïssé limit of the class of finite graphs
+    $\cG$ denoted by $\FrGr = \Flim(\cG)$.
+  \end{definition}
+
+  The concept of the random graph emerges independently in many fields of
+  mathematics. For example, one can construct the graph by choosing at random
+  for each pair of vertices if they should be connected or not. It turns out
+  that the graph constructed this way is isomorphic to the random graph with
+  probability 1.
+
+  The random graph $\FrGr$ has one particular property that is unique to the
+  random graph.
+
+  \begin{fact}[random graph property]
+	For each finite disjoint $X, Y\subseteq \FrGr$ there exists $v\in\FrGr\setminus (X\cup Y)$
+    such that $\forall u\in X (vEu)$ and $\forall u\in Y (\neg vEu)$.
+  \end{fact}
+  \begin{proof}
+    Take any finite disjoint $X, Y\subseteq\FrGr$. Let $G_{XY}$ be the
+    subgraph of $\FrGr$ induced by the $X\cup Y$. Let $H = G_{XY}\cup \{w\}$,
+    where $w$ is a new vertex that does not appear in $G_{XY}$. Also, $w$ is connected to 
+    all vertices of $G_{XY}$ that come from $X$ and to none of those that come
+    from $Y$. This graph is of course finite, so it is embeddable in $\FrGr$.
+    Without loss of generality assume that this embedding is simply inclusion. 
+    Let $f$ be the partial isomorphism from $X\sqcup Y$ to $H$, with $X$ and
+    $Y$ projected to the part of $H$ that come from $X$ and $Y$ respectively. 
+    By the ultrahomogeneity of $\FrGr$ this isomorphism extends to an automorphism
+    $\sigma\in\Aut(\FrGr)$. Then $v = \sigma^{-1}(w)$ is the vertex we sought. 
+  \end{proof}
+
+  \begin{fact}
+    If a countable graph $G$ has the random graph property, then it is
+    isomorphic to the random graph $\FrGr$.
+  \end{fact}
+  \begin{proof}
+    Enumerate vertices of both graphs: $\FrGr = \{a_1, a_2\ldots\}$ and $G
+    = \{b_1, b_2\ldots\}$.
+    We will construct a chain of partial isomorphisms $f_n\colon \FrGr\to G$
+    such that $\emptyset = f_0\subseteq f_1\subseteq f_2\subseteq\ldots$ and $a_n \in
+    \dom(f_n)$ and $b_n\in\rng(f_n)$.
+
+	Suppose we have $f_n$. We seek $b\in G$ such that $f_n\cup \{\langle
+    a_{n+1}, b\rangle\}$ is a partial isomorphism. 
+    If $a_{n+1}\in\dom{f_n}$, then simply $b = f_n(a_{n+1})$. Otherwise,
+	let $X = \{a\in\FrGr\mid
+    aE_{\FrGr} a_{n+1}\}\cap \dom{f_n}, Y = X^{c}\cap \dom{f_n}$, i.e. $X$ are
+    vertices of $\dom{f_n}$ that are connected with $a_{n+1}$ in $\FrGr$ and
+    $Y$ are those vertices that are not connected with $a_{n+1}$. Let $b$ be
+    a vertex of $G$ that is connected to all vertices of $f_n[X]$ and to none
+    $f_n[Y]$ (it exists by the random graph property). Then $f_n\cup \{\langle
+    a_{n+1}, b\rangle\}$ is a partial isomorphism. We find $a$ for the
+    $b_{n+1}$ in the similar manner, so that $f_{n+1} = f_n\cup \{\langle
+    a_{n+1}, b\rangle, \langle a, b_{n+1}\rangle\}$ is a partial isomorphism.
+
+    Finally, $f = \bigcup^{\infty}_{n=0}f_n$ is an isomorphism between $\FrGr$
+    and $G$. Take any $a, b\in \FrGr$. Then for some big enough $n$ we have
+    that $aE_{\FrGr}b\Leftrightarrow f_n(a)E_{G}f_n(b) \Leftrightarrow f(a)E_{G}f(b)$.
+  \end{proof}
+
+  Using this fact one can show that the graph constructed in the probabilistic
+  manner is in fact isomorphic to the random graph $\FrGr$.
+
+  \begin{definition} We say that a Fraïssé class $\bK$ has \emph{weak
+    Hrushovski property} (\emph{WHP}) if for every $A\in \bK$ and an isomorphism
+	of its substructures $p\colon A\to A$ (also called a partial automorphism of $A$),
+	there is some $B\in \bK$ such 
+    that $p$ can be extended to an automorphism of $B$, i.e.\ there is an 
+    embedding $i\colon A\to B$ and a $\bar p\in \Aut(B)$ such that the following
+    diagram commutes: 
+    \begin{center} 
+      \begin{tikzcd} 
+        B\ar[r,dashed,"\bar p"]&B\\
+        A\ar[u,dashed,"i"]\ar[r,"p"]&A\ar[u,dashed,"i"] 
+      \end{tikzcd} 
+    \end{center}
+  \end{definition}
+
+  \begin{proposition}
+    \label{proposition:finite-graphs-whp}
+    The class of finite graphs $\cG$ has the weak Hrushovski property. 
+  \end{proposition}
+
+  \begin{proof}
+    It may be there some day, but it may not!
+  \end{proof}
+
+  \subsection{Canonical amalgamation}
+
+  \begin{definition}
+	Let $\bK$ be a class finitely generated $L$-structures. We say that
+	$\bK$ has \emph{canonical amalgamation} if for every $C\in\bK$ there
+	is a functor $\otimes_C\colon\Cospan(\bK)\to\Pushout(\bK)$ such that
+	it has the following properties:
+	\begin{itemize}
+	  \item Let $A\leftarrow C\rightarrow B$ be a cospan. Then $\otimes_C$ sends
+		it to a pushout that preserves ``the bottom`` structures and embeddings, i.e.:
+		\begin{center}
+		  \begin{tikzcd}
+			& & 	 							& 		 & A\otimes_C B & \\
+			A & & B  \arrow[r, dashed, "A\otimes_C B"] & A \arrow[ur, dashed] & & B \arrow[ul, dashed] \\
+			& C \arrow[ul] \arrow[ur]  & & & C \arrow[ul] \arrow[ur] &
+		  \end{tikzcd}
+		\end{center}
+
+	  We have deliberately omited names for embeddings of $C$. Of course, 
+	  the functor has to take them into account, but this abuse of notation
+	  is convenient and should not lead into confusion.
+	  \item Let $A\leftarrow C\rightarrow B$, $A'\leftarrow C\rightarrow B'$ be cospans
+		with a natural transformation given by $\alpha\colon A\to A', \beta\colon B\to B',
+		\gamma\colon C\to C$. Then $\otimes_C$ preserves the morphisms of
+		when sending the natural transformation of those cospans to natural
+		transformation of pushouts by adding the 
+		$\delta\colon A\otimes_C B\to A'\otimes_C B'$ morphism:
+
+		\begin{center}
+		  \begin{tikzcd}
+			& A'\otimes_C B' & \\
+			A' \arrow[ur] & & B' \arrow[ul] \\
+			& A\otimes_C B  \ar[uu, dashed, "\delta"] & \\
+			& C \arrow[uul, bend left] \arrow[uur, bend right] & \\
+			A \arrow[uuu, dashed, "\alpha"] \arrow[uur, bend left, crossing over] & & B \arrow[uuu, dashed, "\beta"'] \arrow[uul, bend right, crossing over] \\
+			& C \arrow[ur] \arrow[ul] \arrow[uu, dashed, "\gamma"] & \\ 
+		  \end{tikzcd}
+		\end{center}
+		% \begin{center}
+		%   \begin{tikzcd}
+			% & A \ar[rrr, dashed, "\alpha"] \ar[drr, bend left=20, crossing over] & & & A' \ar[dr] & \\
+			% C \ar[rr, dashed, "\gamma"] \ar[ur] \ar[dr] & & C \ar[rrd, bend right=20] \ar[rru, bend left=20] & A\otimes_C B \ar[rr, dashed, "\delta"] & & A' \otimes_C B' \\
+			% & B \ar[rrr, dashed, "\beta"] \ar[urr, bend right=20, crossing over] & & & B' \ar[ur] & \\
+		%   \end{tikzcd}
+		% \end{center}
+	\end{itemize}
+  \end{definition}
+
+  \begin{theorem}
+	\label{theorem:canonical_amalgamation_thm}
+	Let $\bK$ be a Fraïssé class of $L$-structures with canonical amalgamation. 
+	Then the class $\cH$ of $L$-structures with automorphism is a Fraïssé class.
+  \end{theorem}
+
+  \begin{proof}
+	$\cH$ is obviously countable and has HP. It suffices to show that it
+	has AP (JEP follows by taking $C$ to be the empty structure). Take any
+	$(A,\alpha), (B,\beta), (C,\gamma)\in \cH$ such that $(C,\gamma)$ embeds
+	into $(A,\alpha)$ and $(B,\beta)$. Then $\alpha, \beta, \gamma$ yield
+	an automorphism $\eta$ (as a natural transformation) of a cospan:
+	\begin{center}
+	  \begin{tikzcd}
+		A & & B \\
+		% & C \ar[ur] \ar[ul] & \\
+		A \ar[u, dashed, "\alpha"] & C \ar[ur] \ar[ul] & B \ar[u, dashed, "\beta"'] \\
+		& C \ar[ur] \ar[ul] \ar[u, dashed, "\gamma"] &
+		% (A, \alpha) & & (B, \beta) \\
+		% & (C, \gamma) \ar[ur] \ar[ul] &
+	  \end{tikzcd}
+	\end{center}
+	
+	Then, by the fact \ref{fact:functor_iso}, $\otimes_C(\eta)$ is an automorphism
+	of the pushout diagram:
+	
+	\begin{center}
+	  \begin{tikzcd}
+		& A\otimes_C B \ar[loop above, "\delta"] & \\
+		A \ar[ur] \ar[loop left, "\alpha"]& & B \ar[ul] \ar[loop right, "\beta"]\\
+		& C \ar[ur] \ar[ul] \ar[loop below, "\gamma"] & 
+	  \end{tikzcd}
+	\end{center}
+	
+	TODO: ten diagram nie jest do końca taki jak trzeba, trzeba w zasadzie skopiować
+	ten z definicji kanonicznej amalgamcji. Czy to nie będzie wyglądać źle?
+
+	This means that the morphism $\delta\colon A\otimes_C B\to A\otimes_C B$
+	has to be automorphism. Thus, by the fact that the diagram commutes, 
+	we have the amalgamation of $(A, \alpha)$ and $(B, \beta)$ over $(C,\gamma)$ 
+	in $\cH$.
+  \end{proof}
+
+  \subsection{Graphs with automorphism}
+  The language and theory of unordered graphs is fairly simple. We extend the
+  language by one unary symbol $\sigma$ and interpret it as an arbitrary
+  automorphism on the graph structure. It turns out that the class of such
+  structures is a Fraïssé class.
+
+  \begin{proposition}
+    Let $\cH$ be the class of all finite graphs with an automorphism, i.e.
+    structures in the language $(E, \sigma)$ such that $E$ is a symmetric
+    relation and $\sigma$ is an automorphism on the structure. $\cH$ is
+    a Fraïssé class.
+  \end{proposition}
+  \begin{proof}
+    Countability and HP are obvious, JEP follows by the same argument as in
+    plain graphs. We need to show that the class has the amalgamation property.
+
+    Take any $(A, \alpha), (B, \beta), (C,\gamma)\in\cH$ such that $A$ embeds
+    into $B$ and $C$. Without the loss of generality we may assume that 
+	$B\cap C = A$ and $\alpha\subseteq\beta,\gamma$. 
+	Let $D$ be the amalgamation of $B$ and $C$ over $A$ as in
+    the proof for the plain graphs. We will define the automorphism
+    $\delta\in\Aut(D)$ so it extends $\beta$ and $\gamma$. 
+	We let $\delta\upharpoonright_B = \beta, \delta\upharpoonright_C = \gamma$.
+	Let's check the definition is correct. We have to show that 
+	$(uE_Dv\leftrightarrow \delta(u)E_D\delta(v))$ holds for any $u, v\in
+	D$. We have two cases:
+    \begin{itemize}
+      \item $u, v\in X$, where $X$ is either $B$ or $C$. This case is trivial.
+      \item $u\in B\setminus A, v\in C\setminus A$. Then
+        $\delta(u)=\beta(u)\in B\setminus A$, similarly $\delta(v)=\gamma(v)\in
+        C\setminus A$. This follows from the fact, that $\beta\upharpoonright_A
+        = \alpha$, so for any $w\in A\quad\beta^{-1}(w)=\alpha^{-1}(w)\in A$,
+        similarly for $\gamma$. Thus, from the construction of $D$, $\neg uE_Dv$
+        and $\neg \delta(u)E_D\delta(v)$.
+    \end{itemize}
+  \end{proof}
+
+  The proposition says that there is a Fraïssé for the class $\cH$ of finite
+  graphs with automorphisms. We shall call it $(\FrAut, \sigma)$. Not
+  surprisingly, $\FrAut$ is in fact a random graph.
+
+  \begin{proposition}
+	\label{proposition:graph-aut-is-normal}
+	The Fraïssé limit of $\cH$ interpreted as a plain graph (i.e. as a reduct
+	to the language of graphs) is isomorphic to the random graph $\FrGr$. 
+  \end{proposition}
+
+  \begin{proof}
+    It is enough to show that $\FrAut = \Flim(\cH)$ has the random graph
+    property. Take any finite disjoint $X, Y\subseteq \FrAut$. Without the loss
+    of generality assume that $X\cup Y$ is $\sigma$-invariant, i.e.
+    $\sigma(v)\in X\cup Y$ for $v\in X\cup Y$. This assumption can be made 
+    because there are no infinite orbits in $\sigma$, which in turn is due to
+    the fact that $\cH$ is the age of $\FrAut$. 
+
+    Let $G_{XY}$ be the graph induced by $X\cup Y$. Take $H=G_{XY}\sqcup {v}$ 
+    as a supergraph of $G_{XY}$ with one new vertex $v$ connected to all 
+    vertices of $X$ and to none of $Y$. By the proposition
+    \ref{proposition:finite-graphs-whp} we can extend $H$ together with its
+    partial isomorphism $\sigma\upharpoonright_{X\cup Y}$ to a graph $R$ with
+    automorphism $\tau$. Once again, without the loss of generality we can
+    assume that $R\subseteq\FrAut$, because $\cH$ is the age of $\FrAut$. But
+    $R\upharpoonright_{G_{XY}}$ together with $\tau\upharpoonright_{G_{XY}}$
+    are isomorphic to the $G_{XY}$ with $\sigma\upharpoonright_{G_{XY}}$.
+
+    Thus, by ultrahomogeneity of $\FrAut$ this isomorphism extends to an 
+    automorphism $\theta$ of $(\FrAut, \sigma)$. Then $\theta(v)$ is the vertex
+    that is connected to all vertices of $X$ and none of $Y$, because
+    $\theta[R\upharpoonright_X] = X, \theta[R\upharpoonright_Y] = Y$.
+  \end{proof}
+
+
+  \begin{theorem}
+	\label{theorem:isomorphic_fr_lims}
+	Let $\cC$ be a Fraïssé class of finite structures in a relational language
+	$L$ of some theory $T$. Let $\cD$ be a class of finite structures of the
+	theory $T$ in a relational language $L$ with additional unary function 
+	symbol interpreted as an automorphism of the structure. If $\cC$ has the
+	weak Hrushovski property and $\cD$ is a Fraïssé class then the Fraïssé
+	limit of $\cC$ is isomorphic to the Fraïssé limit of $\cD$ reduced
+	to the language $L$.
+  \end{theorem}
+
+  \begin{proof}
+	Let $\Gamma=\Flim(\cC)$ and $(\Pi, \sigma) =\Flim(\cD)$. By the Fraïssé 
+	theorem \ref{theorem:fraisse_thm} it suffices to show that the age of $\Pi$
+	is $\cC$ and that it has the weak ultrahomogeneity in the class $\cC$. The
+	former comes easily, as for every structure $A\in \cC$ we have the structure
+	$(A, \id_A)\in \cD$, which means that the structure $A$ embedds into $\Pi$.
+	Also, if a structure $(B, \beta)\in\cD$ embedds into $\cD$, then $B\in\cC$.
+	Hence, $\cC$ is indeed the age of $\Pi$.
+
+	Now, take any structure $A, B\in\cC$ such that $A\subseteq B$. Without the
+	loss of generality assume that $A = B\cap \Pi$. Let $\bar{A}$ be the
+	smallest structure closed on the automorphism $\sigma$ and containg $A$.
+	It is finite, as $\cC$ is the age of $\Pi$. By the weak Hrushovski property,
+	of $\cC$ let $(\bar{B}, \beta)$ be a structure extending 
+	$(B\cup \bar{A}, \sigma\upharpoonright_{\bar{A}})$. Again, we may assume 
+	that $B\cup \bar{A}\subseteq \bar{B}$. Then, by the fact that $\Pi$ is a 
+	Fraïssé limit of $\cD$ there is an embedding 
+	$f\colon(\bar{B}, \beta)\to(\Pi, \sigma)$
+	such that the following diagram commutes:
+
+    \begin{center}
+      \begin{tikzcd}
+		(A, \emptyset) \arrow[d, "\subseteq"'] \arrow[r, "\subseteq"] & (\bar{A}, \sigma\upharpoonright_A) \arrow[d, "\subseteq"] \arrow[r, "\subseteq"] & (\Pi, \sigma) \\
+		(B, \sigma\upharpoonright_B) \arrow[r, dashed, "\subseteq"'] & (\bar{B}, \beta) \arrow[ur, dashed, "f"]
+      \end{tikzcd}
+    \end{center}
+
+	Then we simply get the following diagram:
+
+	\begin{center}
+	  \begin{tikzcd}
+		A \arrow[d, "\subseteq"'] \arrow[r, "\subseteq"] & \Pi \\
+		B \arrow[ur, dashed, "f\upharpoonright_B"'] 
+	  \end{tikzcd}
+	\end{center}
+
+	which proves that $\Pi$ is indeed a weakly ultrahomogeneous structure in $\cC$.
+	Hence, it is isomorphic to $\Gamma$.
+  \end{proof}
+\end{document}
-- 
cgit v1.2.3

