Fixes

2 files changed, 44 insertions, 33 deletions

diff --git a/sections/conj_classes.tex b/sections/conj_classes.tex
index 87fa416..e795217 100644
--- a/sections/conj_classes.tex
+++ b/sections/conj_classes.tex
@@ -85,9 +85,10 @@
 	Let $\cC$ be a Fraïssé class of finitely generated $L$-structures. 
 	Let $\cD$ be the class of structures from $\cC$ 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 there is a comeagre conjugacy class in the
-	automorphism group of the $\Flim(\cC)$.
+	as an automorphism of the structure. If $\cC$ has the weak Hrushovski 
+	property, $\cD$ is a Fraïssé class and $(\Pi, \sigma) = \Flim(\cD)$, then
+	$\Aut(\cC)$ has a comeagre conjugacy class and $\sigma$ is in this class
+	(i.e. $\sigma$ is a generic automorphism).
   \end{theorem}
 
   Before we get to the proof, it is important to mention that an isomorphism 
@@ -97,7 +98,7 @@
   of finitely generated structures (if it yields one) and \textit{vice versa}. 
 
   \begin{proof}
-	Let $\Gamma = \Flim(\cC)$ and $(\Pi, \sigma) = \Flim(\cD)$. First, by the Theorem
+	Let $\Gamma = \Flim(\cC)$. 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 $\Gamma$.
@@ -228,7 +229,10 @@
   \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.
+	and canonical amalgamation. Let $\cD$ be the class of structures from $\cC$ with 
+	additional unary function symbol interpreted as an automorphism of the structure.
+	Then $\cD$ is a Fraïssé class and for $(\Pi, \sigma) = \Flim(\cD)$ 
+	we have that the conjugacy class of $\sigma$ is comeagre in $\Aut(\cC)$.
   \end{theorem}
 
   \begin{proof}
diff --git a/sections/examples.tex b/sections/examples.tex
index fdb8c6a..3d276ff 100644
--- a/sections/examples.tex
+++ b/sections/examples.tex
@@ -1,6 +1,41 @@
 \documentclass[../lic_malinka.tex]{subfiles}
 
 \begin{document}
+  In this section we give examples and anti-examples of Fraïssé classes
+  with WHP or CAP.
+
+
+  \begin{example}
+	The class of all finite graphs $\cG$ is a Fraïssé class with WHP and free
+	amalgamation.
+  \end{example}
+
+  We have already shown this fact. Thus get that the random graph has a generic
+  automorphism.
+
+  \begin{example}
+	A $K_n$-free graph is a graph with no $n$-clique as its subgraph. 
+	Let $\cG_n$ be the class of finite \emph{$K_n$-free} graphs. $\cG_n$ 
+	is a Fraïssé class with WHP and free amalgamation.
+  \end{example}
+
+  Showing that $\cG_n$ is indeed a Fraïssé class is almost the same as in
+  normal graphs, together with free amalgamation. WHP is trickier and the proof
+  can be seen in \cite{eppa_presentation} Theorem 3.6. Hence, $\Flim(\cG_n)$
+  has a generic automorphism.
+
+  \begin{example}
+	The class $\cV$ of all finitely generated vector spaces over a countable field
+	is a Fraïssé class with WHP and CAP.
+  \end{example}
+
+  Vector spaces of the same dimension are isomorphic, thus it is obvious that
+  $\cV$ is essentially countable. Also $HP$ and $JEP$ are obvious, as we can
+  always embed space with smaller dimension into the bigger one. Amalgamation
+  works exactly the same. In fact, such amalgamation is indeed canonical.
+
+  Now we give some anti-examples:
+
   \begin{example}
 	Let $\cL$ be the class of all finite linear orderings. Then:
 	\begin{enumerate}
@@ -155,32 +190,4 @@
   Thus there cannot be a dense conjugacy class in $\Aut(\Sigma)$ and so there's
   no generic automorphism.
 
-  \begin{example}
-	The class $\cV$ of all finitely generated vector spaces over a countable field
-	is a Fraïssé class with WHP and CAP.
-  \end{example}
-
-  Vector spaces of the same dimension are isomorphic, thus it is obvious that
-  $\cV$ is essentially countable. Also $HP$ and $JEP$ are obvious, as we can
-  always embed space with smaller dimension into the bigger one. Amalgamation
-  works exactly the same. In fact, such amalgamation is indeed canonical.
-
-  \begin{example}
-	The class of all finite graphs $\cG$ is a Fraïssé class with WHP and free
-	amalgamation.
-  \end{example}
-
-  We have already shown this fact. Thus get that the random graph has a generic
-  automorphism.
-
-  \begin{example}
-	A $K_n$-free graph is a graph with no $n$-clique as its subgraph. 
-	Let $\cG_n$ be the class of finite \emph{$K_n$-free} graphs. $\cG_n$ 
-	is a Fraïssé class with WHP and free amalgamation.
-  \end{example}
-
-  Showing that $\cG_n$ is indeed a Fraïssé class is almost the same as in
-  normal graphs, together with free amalgamation. WHP is trickier and the proof
-  can be seen in \cite{eppa_presentation} Theorem 3.6. Hence, $\Flim(\cG_n)$
-  has a generic automorphism.
 \end{document}