More general proposition 3.18

1 files changed, 83 insertions, 2 deletions

diff --git a/lic_malinka.tex b/lic_malinka.tex
index 5e52034..9b218ee 100644
--- a/lic_malinka.tex
+++ b/lic_malinka.tex
@@ -47,6 +47,7 @@
 

 \newcommand{\cupdot}{\mathbin{\mathaccent\cdot\cup}} 

 \newcommand{\cC}{\mathcal C} 

+\newcommand{\cD}{\mathcal D} 

 \newcommand{\cV}{\mathcal{V}} 

 \newcommand{\cU}{\mathcal{U}}

 \newcommand{\cG}{\mathcal{G}}

@@ -144,7 +145,8 @@
     }x\}$ is comeagre in $X$.  

   \end{definition}

 

-  \begin{definition} Let $X$ be a nonempty topological space and let

+  \begin{definition} 

+	Let $X$ be a nonempty topological space and let

     $A\subseteq X$. The \emph{Banach-Mazur game of $A$}, denoted as

     $G^{\star\star}(A)$ is defined as follows: Players $I$ and

     $\textit{II}$ take turns in playing nonempty open sets $U_0, V_0,

@@ -646,6 +648,56 @@
     $\theta[R\upharpoonright_X] = X, \theta[R\upharpoonright_Y] = Y$.

   \end{proof}

 

+  \begin{theorem}

+	Let $\cC$ be a Fraïssé class of finite structures in a relational language

+	$L$ of some theory $T$. Let $\cD$ be a class of finite structures of the

+	theory $T$ in a relational language $L$ 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 the Fraïssé

+	limit of $\cC$ is isomorphic to the Fraïssé limit of $\cD$ reduced

+	to the language $L$.

+  \end{theorem}

+

+  \begin{proof}

+	Let $\Gamma=\Flim(\cC)$ and $(\Pi, \sigma) =\Flim(\cD)$. By the Fraïssé 

+	theorem \ref{theorem:fraisse_thm} it suffices to show that the age of $\Pi$

+	is $\cC$ and that it has the weak ultrahomogeneity in the class $\cC$. The

+	former comes easily, as for every structure $A\in \cC$ we have the structure

+	$(A, \id_A)\in \cD$, which means that the structure $A$ embedds into $\Pi$.

+	Also, if a structure $(B, \beta)\in\cD$ embedds into $\cD$, then $B\in\cC$.

+	Hence, $\cC$ is indeed the age of $\Pi$.

+

+	Now, take any structure $A, B\in\cC$ such that $A\subseteq B$. Without the

+	loss of generality assume that $A = B\cap \Pi$. Let $\bar{A}$ be the

+	smallest structure closed on the automorphism $\sigma$ and containg $A$.

+	It is finite, as $\cC$ is the age of $\Pi$. By the weak Hrushovski property,

+	of $\cC$ let $(\bar{B}, \beta)$ be a structure extending 

+	$(B\cup \bar{A}, \sigma\upharpoonright_{\bar{A}})$. Again, we may assume 

+	that $B\cup \bar{A}\subseteq \bar{B}$. Then, by the fact that $\Pi$ is a 

+	Fraïssé limit of $\cD$ there is an embedding 

+	$f\colon(\bar{B}, \beta)\to(\Pi, \sigma)$

+	such that the following diagram commutes:

+

+    \begin{center}

+      \begin{tikzcd}

+		(A, \emptyset) \arrow[d, "\subseteq"'] \arrow[r, "\subseteq"] & (\bar{A}, \sigma\upharpoonright_A) \arrow[d, "\subseteq"] \arrow[r, "\subseteq"] & (\Pi, \sigma) \\

+		(B, \sigma\upharpoonright_B) \arrow[r, dashed, "\subseteq"'] & (\bar{B}, \beta) \arrow[ur, dashed, "f"]

+      \end{tikzcd}

+    \end{center}

+

+	Then we simply get the following diagram:

+

+	\begin{center}

+	  \begin{tikzcd}

+		A \arrow[d, "\subseteq"'] \arrow[r, "\subseteq"] & \Pi \\

+		B \arrow[ur, dashed, "f\upharpoonright_B"'] 

+	  \end{tikzcd}

+	\end{center}

+

+	which proves that $\Pi$ is indeed a weakly ultrahomogeneous structure in $\cC$.

+	Hence, it is isomorphic to $\Gamma$.

+  \end{proof}

+

   \section{Conjugacy classes in automorphism groups}

 

   TODO:

@@ -659,6 +711,7 @@
   $[f]_{G} = \{g\in G\mid f\subseteq g\}$.  

 

   \subsection{Prototype: pure set} 

+

   In this section, $M=(M,=)$ is an infinite countable set (with no structure

   beyond equality).

   

@@ -881,7 +934,8 @@
 

 	Take any finite graph with automorphism $(B, \beta)$. Then, there are

 	$i, j$ such that $(B, \beta) = (B_{i, j}, \beta_{i,j})$ and $A_{i,j}=\emptyset$.

-	By the bookkeeping there was $n$ such that $(i, j) = \min\{\{0,1,\ldots\}\times\bN\setminus X_{n-1}\}$.

+	By the bookkeeping there was $n$ such that 

+	$(i, j) = \min\{\{0,1,\ldots\}\times\bN\setminus X_{n-1}\}$.

 	This means that $(B, \beta)$ embeds into $(\FrGr_n, g_n)$, hence it embeds

 	into $(\FrGr, g)$, thus it has age $\cH$. 

 	With a similar argument we can see that $(\FrGr, g)$ is	weakly ultrahomogeneous.

@@ -893,5 +947,32 @@
 	set $A$.

   \end{proof}

 

+  \begin{corollary}

+	Let $\mathcal{W}$ be a Fraïssé class of finitely generated $L$-structures of 

+	a theory $T$. Let $\mathcal{V}$ be the class of finitely generated structures

+	of $T$ with an additional unary function interpreted as an automoprphism of

+	the structure. If $\mathcal{W}$ has weak Hrushovski property and $\mathcal{V}$

+	is a Fraïssé class, then $\mathcal{W}$ has a generic automorphism.

+  \end{corollary}

+

+  TODO: pokazać że w ogólności granica Fraissego V bez tego automorfizmu jest

+  izomorficzna z W, dopiero wtedy można ten dowód tak uogólnić.

+

+  \begin{proof}

+	The proof is an abstract version of the theorem for the random graph.

+  \end{proof}

+

+  \subsection{Properties of the generic automorphism}

+

+  \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}

   \printbibliography

 \end{document}