From 22dfe4dade829c5b9fc1f8928f0c78ce99084e19 Mon Sep 17 00:00:00 2001
From: Franciszek Malinka <franciszek.malinka@gmail.com>
Date: Sun, 29 May 2022 15:20:55 +0200
Subject: Kluczowy dowod uogolniony!

---
 lic_malinka.tex | 153 +++++++++++++++++++++-----------------------------------
 1 file changed, 57 insertions(+), 96 deletions(-)

(limited to 'lic_malinka.tex')

diff --git a/lic_malinka.tex b/lic_malinka.tex
index 9b218ee..9cce430 100644
--- a/lic_malinka.tex
+++ b/lic_malinka.tex
@@ -649,6 +649,7 @@
   \end{proof}
 
   \begin{theorem}
+	\label{theorem:isomorphic_fr_lims}
 	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 
@@ -779,35 +780,32 @@
     $g\circ f_1 = f_2 \circ g$ which exactly means that $f_1, f_2$ conjugate.
   \end{proof} 
 
-
-  % \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}
-
   \begin{theorem}
-	\label{theorem:generic_aut_graph}
-	Let $\FrGr$ be the Fraïssé limit of the class of all finite graphs $\bK$.
-	Then $\FrGr$ has a generic automorphism $\tau\in\Aut(\FrGr)$, i.e. the
-	conjugacy class of $\tau$ is comeagre in $G = \Aut(\FrGr)$.
+	\label{theorem:generic_aut_general}
+	Let $\cC$ be a Fraïssé class of finite structures of a theory $T$ in a 
+	relational language $L$. Let $\cD$ be the class of the finite structures of
+	$T$ in the 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 there is a comeagre conjugacy class in the
+	automorphism group of the $\Flim(\cC)$.
   \end{theorem}
 
   \begin{proof}
+	Let $\Gamma = \Flim(\cC)$ and $(\Pi, \sigma) = \Flim(\cD)$. Let $G = \Aut(\Gamma)$,
+	i.e. $G$ is the automorphism group of $\Gamma$. First, by the theorem
+	\ref{theorem:isomorphic_fr_lims}, we may assume without the loss of generality
+	that $\Pi = \Gamma$.
 	We will construct a strategy for the second player in the Banach-Mazur game
 	on the topological space $G$. This strategy will give us a subset 
-	$A\subseteq G$ and as we will see, this will also be a subset of the 
-	conjugacy class of $\tau$. By the Banach-Mazur theorem 
-	\ref{theorem:banach_mazur_thm} this will prove that the class is comeagre.
+	$A\subseteq G$ and as we will see a subset of a cojugacy class in $G$.
+	By the Banach-Mazur theorem \ref{theorem:banach_mazur_thm} this will prove 
+	that this class is comeagre.
 
 	Recall, $G$ has a basis consisting of open
 	sets $\{g\in G\mid g\upharpoonright_A = g_0\upharpoonright_A\}$ for some
-	finite set $A\subseteq \FrGr$ and some automorphism $g_0\in G$. In other
+	finite set $A\subseteq \Gamma$ and some automorphism $g_0\in G$. In other
 	words, a basic open set is a set of all extensions of some finite partial
-	isomorphism $g_0$ of $\FrGr$. By $B_{g}\subseteq G$ we denote a basic 
+	isomorphism $g_0$ of $\Gamma$. By $B_{g}\subseteq G$ we denote a basic 
 	open subset given by a finite partial isomorphism $g$. From now on we will
 	consider only finite partial isomorphism $g$ such that $B_g$ is nonempty.
 
@@ -819,24 +817,22 @@
 	the corresponding basic open subsets $B_{f_0}\supseteq B_{g_0}\supseteq\ldots$.
 
 	Our goal is to choose $g_i$ in such a manner that the resulting function
-	$g = \bigcap^{\infty}_{i=0}g_i$ will be an automorphism of the random graph
-	such that $(\FrGr, g) = \Flim{\cH}$, i.e. the Fraïssé limit of finite
-	graphs with automorphism. Precisely, $\bigcap^{\infty}_{i=0}B_{g_i} = \{g\}$.
-	By the Fraïssé theorem \ref{theorem:fraisse_thm}
-	it will follow that $(\FrGr, g)\cong (\FrAut, \sigma)$. By the 
-	proposition \ref{proposition:graph-aut-is-normal} we can assume without
-	the loss of generality that $\FrAut = \FrGr$ as a plain graph. Hence,
+	$g = \bigcap^{\infty}_{i=0}g_i$ will be an automorphism of the Fraïssé limit 
+	$\Gamma$ such that $(\Gamma, g) = \Flim{\cD}$. 
+	Precisely, $\bigcap^{\infty}_{i=0}B_{g_i} = \{g\}$, 
+	by the Fraïssé theorem \ref{theorem:fraisse_thm}
+	it will follow that $(\Gamma, g)\cong (\Pi, \sigma)$. Hence,
 	by the fact	\ref{fact:conjugacy}, $g$ and $\sigma$ conjugate in $G$.
 
 	Once again, by the Fraïssé theorem \ref{theorem:fraisse_thm} and the
 	\ref{lemma:weak_ultrahom} lemma constructing $g_i$'s in a way such that
-	age of $(\FrGr, g)$ is exactly $\cH$ and so that it is weakly ultrahomogeneous
-	will produce expected result. First, let us enumerate all pairs of finite
-	graphs with automorphism $\{\langle(A_n, \alpha_n), (B_n, \beta_n)\rangle\}_{n\in\bN}$
+	age of $(\Gamma, g)$ is exactly $\cD$ and so that it is weakly ultrahomogeneous
+	will produce expected result. First, let us enumerate all pairs of structures
+	$\{\langle(A_n, \alpha_n), (B_n, \beta_n)\rangle\}_{n\in\bN},\in\cD$
 	such that the first element of the pair embeds by inclusion in the second,
 	i.e. $(A_n, \alpha_n)\subseteq (B_n, \beta_n)$. Also, it may be that 
-	$A_n$ is an empty graph. We enumerate the vertices of the random graph 
-	$\FrGr = \{v_0, v_1, \ldots\}$.
+	$A_n$ is an empty. We enumerate the elements of the Fraïssé limit 
+	$\Gamma = \{v_0, v_1, \ldots\}$.
 
 	Fix a bijection $\gamma\colon\bN\times\bN\to\bN$ such that for any
 	$n, m\in\bN$ we have $\gamma(n, m) \ge n$. This bijection naturally induces
@@ -850,15 +846,15 @@
 	such that following properties hold:
 
 	\begin{enumerate}[label=(\roman*)]
-	  \item $g_n$ is an automorphism of the induced subgraph $\FrGr_n$,
+	  \item $g_n$ is an automorphism of the induced substructure $\Gamma_n$,
 	  \item $g_n(v_n)$ and $g_n^{-1}(v_n)$ are defined,
 	  \item let 
 		$\{\langle (A_{n,k}, \alpha_{n, k}), (B_{n,k}, \beta_{n,k}), f_{n, k}\rangle\}_{k\in\bN}$
-		be the enumeration of all pairs of finite graphs with automorphism such
-		that the first is a substructure of the second, i.e. 
+		be the enumeration of all pairs of finite structures of $T$ with automorphism 
+		such that the first is a substructure of the second, i.e. 
 		$(A_{n,k}, \alpha_{n,k})\subseteq (B_{n,k}, \beta_{n,k})$, and $f_{n,k}$
 		is an embedding	of $(A_{n,k}, \alpha_{n,k})$ in the $\FrGr_{n-1}$ (which
-		is the graph induced by $g_{n-1}$). Let 
+		is the substructure induced by $g_{n-1}$). Let 
 		$(i, j) = \min\{(\{0, 1, \ldots\} \times \bN) \setminus X_{n-1}\}$ (with the
 		order induced by $\gamma$). Then $X_n = X_{n-1}\cup\{(i,j)\}$ and
 		$(B_{n,k}, \beta_{n,k})$ embeds in $(\FrGr_n, g_n)$ so that this diagram
@@ -866,79 +862,59 @@
 
 		\begin{center}
 		  \begin{tikzcd}
-									  & (\FrGr_n, g_n) & \\
+									  & (\Gamma_n, g_n) & \\
 			(B_{i,j}, \beta_{i,j}) \arrow[ur, dashed, "\hat{f}_{i,j}"] &   & (\FrGr_{n-1}, g_{n-1}) \arrow[ul, dashed, "\subseteq"'] \\
 									  & (A_{i,j}, \alpha_{i,j}) \arrow[ur, "f_{i,j}"'] \arrow[ul, "\subseteq"]
 		  \end{tikzcd}
 		\end{center}
-
-		% \begin{center}
-		%   \begin{tikzcd}
-		% 	(A_{i,j}, \alpha_{i,j}) \arrow[d, "\subseteq"'] \arrow[r, "f_{i,j}"] & (\FrGr_{n-1}, g_{n-1}) \arrow[d, "\subseteq"] \\
-		% 	(B_{i,j}, \beta_{i,j}) \arrow[r, dashed, "\hat{f}_{i,j}"'] & (\FrGr_n, g_n)
-		%   \end{tikzcd}
-		% \end{center}
 	\end{enumerate}
 
-	First item makes sure that no infinite orbit will not be present in $g$. The 
+	First item makes sure that no infinite orbit will be present in $g$. The 
 	second item together with the first one are necessary for $g$ to be an 
-	automorphism of $\FrGr$. The third item is the one that gives weak
+	automorphism of $\Gamma$. The third item is the one that gives weak
 	ultrahomogeneity. Now we will show that indeed such $g_n$ may be constructed.
 	
-	First, we will suffice the item (iii). Namely, we will construct $\FrGr'_n, g'_n$
+	First, we will suffice the item (iii). Namely, we will construct $\Gamma'_n, g'_n$
 	such that $g_{n-1}\subseteq g'_n$ and $f_{i,j}$ extends to an embedding of
-	$(B_{i,j}, \beta_{i,j})$ to $(\FrGr'_n, g'_n)$. But this can be easily
-	done by the fact, that $\cH$ has the amalgamation property. Moreover, without
+	$(B_{i,j}, \beta_{i,j})$ to $(\Gamma'_n, g'_n)$. But this can be easily
+	done by the fact, that $\cD$ has the amalgamation property. Moreover, without
 	the loss of generality we can assume that all embeddings are inclusions.
 
     \begin{center}
       \begin{tikzcd}
-                                  & (\FrGr'_n, g'_n) & \\
-        (B_{i,j}, \beta_{i,j}) \arrow[ur, dashed, "\subseteq"] &   & (\FrGr_{n-1}, g_{n-1}) \arrow[ul, dashed, "\subseteq"'] \\
+                                  & (\Gamma'_n, g'_n) & \\
+        (B_{i,j}, \beta_{i,j}) \arrow[ur, dashed, "\subseteq"] &   & (\Gamma_{n-1}, g_{n-1}) \arrow[ul, dashed, "\subseteq"'] \\
 								  & (A_{i,j}, \alpha_{i,j}) \arrow[ur, "\subseteq"'] \arrow[ul, "\subseteq"]
       \end{tikzcd}
     \end{center}
 
-	% Without the loss of generality
-	% we may assume that $f_{i,j}$ is an inclusion and that $A_{i,j} = B_{i,j}\cap\FrGr_{n-1}$.
-	% Then let $\FrGr'_n = B_{i,j}\cup\FrGr_{n-1}$ and $g'_n = g_{n-1}\cup \beta_{i,j}$.
-	% Then $(B_{i,j}, \beta_{i,j})$ simply embeds by inclusion in $(\FrGr'_n, g'_n)$,
-	% i.e. this diagram commutes:
-
-	% \begin{center}
-	%   \begin{tikzcd}
-	% 	(A_{i,j}, \alpha_{i,j}) \arrow[d, "\subseteq"'] \arrow[r, "\subseteq"] & (\FrGr_{n-1}, g_{n-1}) \arrow[d, "\subseteq"] \\
-	% 	(B_{i,j}, \beta_{i,j}) \arrow[r, dashed, "\subseteq"'] & (\FrGr'_n, g'_n)
-	%   \end{tikzcd}
-	% \end{center}
-
-	% The argument that those are actually embeddings is almost the same as in
-	% proof of the amalgamation property of $\cH$.	
-
-	It may be also assumed without the loss of generality that $\FrGr'_n\subseteq \FrGr$.
-	Of course by the recursive assumption $\FrGr_{n-1}\subseteq\FrGr$. The 
-	$\FrGr'_n \setminus\FrGr_{n-1} = B_{i,j}\setminus A_{i,j}$ can be found in
-	$\FrGr$ by the random graph property -- we can find vertices of the remaining
-	part of $B_{i,j}$ each at a time so that all edges are correct.
-
-	Now, by the WHP of $\bK$ we can extend the graph $\FrGr'_n\cup\{v_n\}$ together
-	with its partial isomorphism $g'_n$ to a graph $\FrGr_n$ together with its
+	By the weak ultrahomogeneity we may assume that $\Gamma'_n\subseteq \Gamma$:
+
+	\begin{center}
+	  \begin{tikzcd}
+		(B_{i,j}\cup\Gamma_{n-1}, \beta_{i,j}\cup g_{n-1}) \arrow[d, "\subseteq"'] \arrow[r, "\subseteq"] & \Gamma \\
+		(\Gamma'_{n}, g'_n)\arrow[ur, dashed, "f"'] 
+	  \end{tikzcd}
+	\end{center}
+
+	Now, by the WHP of $\bK$ we can extend the graph $\Gamma'_n\cup\{v_n\}$ together
+	with its partial isomorphism $g'_n$ to a graph $\Gamma_n$ together with its
 	automorphism $g_n\supseteq g'_n$ and without the loss of generality we
-	may assume that $\FrGr_n\subseteq\FrGr$. This way we've constructed $g_n$
+	may assume that $\Gamma_n\subseteq\Gamma$. This way we've constructed $g_n$
 	that has all desired properties.
 
 	Now we need to see that $g = \bigcap^{\infty}_{n=0}g_n$ is indeed an automorphism
-	of $\FrGr$ such that $(\FrGr, g)$ has the age $\cH$ and is weakly ultrahomogeneous.
-	It is of course an automorphism of $\FrGr$ as it is defined for every $v\in\FrGr$
+	of $\Gamma$ such that $(\Gamma, g)$ has the age $\cH$ and is weakly ultrahomogeneous.
+	It is of course an automorphism of $\Gamma$ as it is defined for every $v\in\Gamma$
 	and is a sum of increasing chain of finite automorphisms.
 
-	Take any finite graph with automorphism $(B, \beta)$. Then, there are
+	Take any finite structure of $T$ 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 
 	$(i, j) = \min\{\{0,1,\ldots\}\times\bN\setminus X_{n-1}\}$.
-	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.
+	This means that $(B, \beta)$ embeds into $(\Gamma_n, g_n)$, hence it embeds
+	into $(\Gamma, g)$, thus it has age $\cH$. 
+	With a similar argument we can see that $(\Gamma, g)$ is weakly ultrahomogeneous.
 
 	By this we know that $g$ and $\sigma$ conjugate in $G$. As we stated in the
 	beginning of the proof, the set $A$ of possible outcomes of the game (i.e.
@@ -947,21 +923,6 @@
 	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}
-- 
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