From e0fea5a2063f6babc496f593dbb05f8814bddc67 Mon Sep 17 00:00:00 2001
From: Franciszek Malinka <franciszek.malinka@gmail.com>
Date: Sun, 8 May 2022 19:42:41 +0200
Subject: More general proposition 3.18

---
 lic_malinka.tex | 85 +++++++++++++++++++++++++++++++++++++++++++++++++++++++--
 1 file changed, 83 insertions(+), 2 deletions(-)

(limited to 'lic_malinka.tex')

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}
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