From b3dab8fb10581feca94a76364b2ed4298675dbf8 Mon Sep 17 00:00:00 2001
From: Franciszek Malinka <franciszek.malinka@gmail.com>
Date: Sun, 10 Jul 2022 17:24:08 +0200
Subject: Theorem 3.23 fixed (hopefully)

---
 sections/fraisse_classes.tex | 68 ++++++++++++++++++++++++++++++++------------
 1 file changed, 50 insertions(+), 18 deletions(-)

(limited to 'sections/fraisse_classes.tex')

diff --git a/sections/fraisse_classes.tex b/sections/fraisse_classes.tex
index 32804f2..5e45400 100644
--- a/sections/fraisse_classes.tex
+++ b/sections/fraisse_classes.tex
@@ -486,32 +486,64 @@
 	Now, take any structures $A, B\in\cC$ such that $A\subseteq B$. Without the
 	loss of generality assume that $A = B\cap \Pi$. Let $\bar{A}\subseteq\Pi$ 
 	be the
-	smallest structure closed under the automorphism $\sigma$ and containing $A$.
-	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:
+	smallest substructure closed under the automorphism 
+	$\sigma$ and containing $A$. It is finitely generated, as $\cC$ is the age 
+	of $\Pi$. 
+	Let $C$ be a finitely generated structure such that 
+	$\bar{A}\rightarrow C \leftarrow B$. Such structure exists by the JEP
+	of $\cC$. Again, we may assume without the loss of generality that
+	$\bar{A}\subseteq C$. Then $\sigma\upharpoonright_{\bar{A}}$ is a
+	partial isomorphism of $C$, hence by the WHP it can be extended to
+	a structure $(\bar{C}, \gamma)\in\cD$ such that 
+	$\gamma\upharpoonright_{\bar{A}} = \sigma\upharpoonright_{\bar{A}}$.
+
+	Then, by the weak ultrahomogeneity of $(\Pi, \sigma)$ we can find an
+	embedding $g$ of $(\bar{C},\gamma)$ such that the following diagram commutes:
+
+	\begin{center}
+	  \begin{tikzcd}
+		(\bar{A}, \sigma\upharpoonright_{\bar{A}}) \ar[d, "\subseteq"] \ar[r, "\subseteq"] & (\Pi, \sigma) \\
+		(\bar{C}, \gamma) \ar[ur, "g"'] &
+	  \end{tikzcd}
+	\end{center}
 
-    \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:
+	Thus, we have that the following diagram commutes:
 
 	\begin{center}
 	  \begin{tikzcd}
-		A \arrow[d, "\subseteq"'] \arrow[r, "\subseteq"] & \Pi \\
-		B \arrow[ur, dashed, "f\upharpoonright_B"'] 
+		A \ar[r, "\subseteq"] \ar[d, "\subseteq"] & \bar{A} \ar[r, "\subseteq"] \ar[d, "\subseteq"] & \Pi \\
+		B \ar[r, "f"] & C \ar[r, "\subseteq"] & \bar{C} \ar[u, "g"] \\
 	  \end{tikzcd}
 	\end{center}
 
-	which proves that $\Pi$ is indeed a weakly ultrahomogeneous structure in $\cC$.
+	%
+	% By the weak Hrushovski property
+	% of $\cC$ let $(\bar{B}, \beta)$ be a structure extending 
+	% $(B, \sigma\upharpoonright_{A})$. 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.
 	Hence, it is isomorphic to $\Gamma$.
   \end{proof}
 \end{document}
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