Fixed prop 4.6

1 files changed, 32 insertions, 10 deletions

diff --git a/sections/conj_classes.tex b/sections/conj_classes.tex
index e795217..c72fd38 100644
--- a/sections/conj_classes.tex
+++ b/sections/conj_classes.tex
@@ -98,7 +98,7 @@
   of finitely generated structures (if it yields one) and \textit{vice versa}. 
 
   \begin{proof}
-	Let $\Gamma = \Flim(\cC)$. First, by the Theorem
+	Let $\Gamma = \Flim(\cC)$ and $(\Pi, \sigma) = \Flim(\cD)$. First, by the Theorem
 	\ref{theorem:isomorphic_fr_lims}, we may assume without the loss of generality
 	that $\Pi = \Gamma$. Let $G = \Aut(\Gamma)$,
 	i.e. $G$ is the automorphism group of $\Gamma$.
@@ -245,7 +245,7 @@
   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 structure.
+  of the structures of $\cC$ with additional automorphism of the structure).
   Let $\Gamma = \Flim(\cC)$. 
 
   \begin{proposition}
@@ -255,13 +255,35 @@
   \end{proposition}
 
   \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 $\cD$, 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 $\cD$.
+	Let $S = \{x\in \Gamma\mid \sigma(x) = x\}$. It is a substructure of $\Gamma$,
+	as it is closed under operations. For any $n$-tuple $a_1,\ldots,a_n\in S$
+	and $n$-ary function symbol $f$ we have that 
+	$f(a_1,\ldots a_n) = f(\sigma(a_1),\ldots, \sigma(a_n)) = \sigma(f(a_1,\ldots,a_n)$,
+	which exactly says that $f(a_1,\ldots,a_n)$ is a fixed point. Also, constant
+	symbols have to be fixed under $\sigma$.
+
+	It is obvious that the
+	age of $S$ is $\cC$, as every structure from $\cC$ with identity embeds
+	into $\Gamma$, and hence into $S$. Also it is weakly ultrahomogeneous.
+	Take $A\subseteq B$ with embedding $f\colon A\to S$. It can be thought
+	as embedding of $(A, \id_A)$ into $(\Gamma,\sigma)$ and thus by its weak
+	ultrahomogeneity we have $\hat{f}\colon (B,\id_B)\to (\Gamma,\sigma)$ and
+	hence $\hat{f}$ is also and embedding of $B$ into $S$ such that the
+	following diagram commutes:
+
+	\begin{center}
+	  \begin{tikzcd}
+		A \ar[r, "f"] \ar[d, "\subseteq"] & S \\
+		B \ar[ru, "\hat{f}"]
+	  \end{tikzcd}
+	\end{center}
+	% First we need to show that it is an infinite substructure of $\Gamma$. 
+	% By the theorem \ref{theorem:generic_aut_general}
+	% 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 $\cD$.
   \end{proof}
 \end{document}