Zmiana kolejności mała

1 files changed, 63 insertions, 62 deletions

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}