From 930d96d3091323e34a9c35dbc58d377666c3311e Mon Sep 17 00:00:00 2001
From: Franciszek Malinka <franciszek.malinka@gmail.com>
Date: Sat, 26 Mar 2022 15:25:46 +0100
Subject: =?UTF-8?q?Zmiana=20kolejno=C5=9Bci=20ma=C5=82a?=
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---
 lic_malinka.tex | 125 ++++++++++++++++++++++++++++----------------------------
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diff --git a/lic_malinka.tex b/lic_malinka.tex
index d98ecee..ede7008 100644
--- a/lic_malinka.tex
+++ b/lic_malinka.tex
@@ -455,67 +455,6 @@
   % $\Flim(\bK)$ is $\aleph_0$-categorical and has quantifier elimination.
   % \end{fact}
 
-  \section{Conjugacy classes in automorphism groups}
-
-  \subsection{Prototype: pure set} 
-  In this section, $M=(M,=)$ is an infinite countable set (with no structure
-  beyond equality).
-  \begin{proposition} 
-    If $f_1,f_2\in \Aut(M)$, then $f_1$ and $f_2$ are conjugate if and only 
-    if for each $n\in \bN\cup \{\aleph_0\}$, $f_1$ and $f_2$ have the same 
-    number of orbits of size $n$.
-  \end{proposition}
-
-  \begin{proposition} The conjugacy class of $f\in \Aut(M)$ is dense if
-    and only if...  \end{proposition} \begin{proposition} If $f\in
-  \Aut(M)$ has an infinite orbit, then the conjugacy class of $f$ is
-  meagre.  
-  \end{proposition}
-
-  % \begin{proposition} 
-    % An automorphism $f$ of $M$ is generic if and only if...
-  % \end{proposition}
-
-  % \begin{proof}
-
-  % \end{proof}
-
-  \subsection{More general structures}
-
-  \begin{fact} 
-    Suppose $M$ is an arbitrary structure and $f_1,f_2\in \Aut(M)$. 
-    Then $f_1$ and $f_2$ are conjugate if and only if $(M,f_1)\cong
-    (M,f_2)$ as structures with one additional unary relation that is
-    an automorphism.  
-  \end{fact}
-
-  \begin{proof}
-    Suppose that $f_1 = g^{-1}f_2g$ for some $g\in \Aut(M)$. 
-    Then $g$ is the automorphism we're looking for. On the other hand if 
-    $g\colon (M, f_1)\to (M, f_2)$ is an isomorphism, then 
-    $g\circ f_1 = f_2 \circ g$ which exactly means that $f_1, f_2$ conjugate.
-  \end{proof} 
-
-  \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 substructures of $A$ $p\colon A\to A$, 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} Suppose $\cC$ is a Fraïssé class in a relational
-  % language with WHP. Then generically, for an $f\in \Aut(\Flim(\cC))$, all
-  % orbits of $f$ are finite.  \end{proposition} \begin{proposition} Suppose
-  %   $\cC$ is a Fraïssé class in an arbitrary countable language with WHP.
-  %   Then generically, for an $f\in \Aut(\Flim(\cC))$ ...  \end{proposition}
-
   \subsection{Random graph} 
   
   In this section we'll take a closer look on a class of finite graphs, which
@@ -605,7 +544,69 @@
     It may be there some day, but it may not!
   \end{proof}
 
-  %   \begin{proposition} Generically, the set of fixed points of $f\in
+  \section{Conjugacy classes in automorphism groups}
+
+  \subsection{Prototype: pure set} 
+  In this section, $M=(M,=)$ is an infinite countable set (with no structure
+  beyond equality).
+  \begin{proposition} 
+    If $f_1,f_2\in \Aut(M)$, then $f_1$ and $f_2$ are conjugate if and only 
+    if for each $n\in \bN\cup \{\aleph_0\}$, $f_1$ and $f_2$ have the same 
+    number of orbits of size $n$.
+  \end{proposition}
+
+  \begin{proposition} The conjugacy class of $f\in \Aut(M)$ is dense if
+    and only if...  \end{proposition} \begin{proposition} If $f\in
+  \Aut(M)$ has an infinite orbit, then the conjugacy class of $f$ is
+  meagre.  
+  \end{proposition}
+
+  % \begin{proposition} 
+    % An automorphism $f$ of $M$ is generic if and only if...
+  % \end{proposition}
+
+  % \begin{proof}
+
+  % \end{proof}
+
+  \subsection{More general structures}
+
+  \begin{fact} 
+    Suppose $M$ is an arbitrary structure and $f_1,f_2\in \Aut(M)$. 
+    Then $f_1$ and $f_2$ are conjugate if and only if $(M,f_1)\cong
+    (M,f_2)$ as structures with one additional unary relation that is
+    an automorphism.  
+  \end{fact}
+
+  \begin{proof}
+    Suppose that $f_1 = g^{-1}f_2g$ for some $g\in \Aut(M)$. 
+    Then $g$ is the automorphism we're looking for. On the other hand if 
+    $g\colon (M, f_1)\to (M, f_2)$ is an isomorphism, then 
+    $g\circ f_1 = f_2 \circ g$ which exactly means that $f_1, f_2$ conjugate.
+  \end{proof} 
+
+  \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 substructures of $A$ $p\colon A\to A$, 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} Suppose $\cC$ is a Fraïssé class in a relational
+  % language with WHP. Then generically, for an $f\in \Aut(\Flim(\cC))$, all
+  % orbits of $f$ are finite.  \end{proposition} \begin{proposition} Suppose
+  %   $\cC$ is a Fraïssé class in an arbitrary countable language with WHP.
+  %   Then generically, for an $f\in \Aut(\Flim(\cC))$ ...  \end{proposition}
+
+    %   \begin{proposition} Generically, the set of fixed points of $f\in
   %   \Aut(M)$ is isomorphic to $M$ (as a graph).  \end{proposition}
+
   \printbibliography
 \end{document}
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