From ad63d0a98d8595d3267fd81196cf8c96361bd911 Mon Sep 17 00:00:00 2001
From: Franciszek Malinka <franciszek.malinka@gmail.com>
Date: Mon, 11 Jul 2022 20:13:42 +0200
Subject: Updates updates

---
 sections/conj_classes.tex | 35 +++++++++++++++++++----------------
 1 file changed, 19 insertions(+), 16 deletions(-)

(limited to 'sections/conj_classes.tex')

diff --git a/sections/conj_classes.tex b/sections/conj_classes.tex
index 26229b0..96522f5 100644
--- a/sections/conj_classes.tex
+++ b/sections/conj_classes.tex
@@ -232,22 +232,25 @@
 	set $A$.
   \end{proof}
 
+  \begin{theorem}
+	\label{theorem:key-theorem}
+	Let $\cC$ be a Fraïssé class of finitely generated $L$-structures with WHP
+	and canonical amalgamation. Then $\Flim(\cC)$ has a generic automorphism.
+  \end{theorem}
+
+  \begin{proof}
+	It follows trivially from Corollary \ref{corollary:whp+canonical-iso}
+	and the above Theorem \ref{theorem:generic_aut_general}.
+  \end{proof}
+
   \subsection{Properties of the generic automorphism}
 
-  Let $\cC$ be a Fraïssé class in a finite relational language $L$ with
-  weak Hrushovski property. Let $\cH$ be the Fraïssé class of the $L$-structures
-  with additional automorphism symbol. Let $\Gamma = \Flim(\cC)$.
-
- %  \begin{proposition}
-	% Let $\sigma$ be the generic automorphism of the random graph $\FrGr$. Then
-	% the graph induced by the set of the fixed points of $\sigma$ is isomorphic
-	% to $\FrGr$.
- %  \end{proposition}
-	%
- %  \begin{proof}
-	% Let $F = \{v\in\FrGr\mid \sigma(v) = v\}$. It suffices to show that $F$ is
-	% infinite and has the random graph property.
- %  \end{proof}
+  Let $\cC$ be a Fraïssé class of finitely generated $L$-structures with
+  weak Hrushovski property and canonical amalgamation. 
+  Let $\cD$ be the Fraïssé class (by the Theorem \ref{theorem:key-theorem}
+  of the structures of $\cC$ with additional automorphism of the strucutre.
+  Let $\Gamma = \Flim(\cC)$. 
+
   \begin{proposition}
 	\label{proposition:fixed_points}
 	Let $\sigma$ be the generic automorphism of $\Gamma$. Then the set
@@ -257,11 +260,11 @@
   \begin{proof}
 	Let $S = \{x\in \Gamma\mid \sigma(x) = x\}$.
 	First we need to show that it is an infinite. By the theorem \ref{theorem:generic_aut_general}
-	we know that $(\Gamma, \sigma)$ is the Fraïssé limit of $\cH$, thus we
+	we know that $(\Gamma, \sigma)$ is the Fraïssé limit of $\cD$, thus we
 	can embed finite $L$-structures of any size with identity as an 
 	automorphism of the	structure into $(\Gamma, \sigma)$. Thus $S$ has to be
 	infinite. Also, the same argument shows that the age of the structure is
 	exactly $\cC$. It is weakly ultrahomogeneous, also by the fact that
-	$(\Gamma, \sigma)$ is in $\cH$.
+	$(\Gamma, \sigma)$ is in $\cD$.
   \end{proof}
 \end{document}
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