Kluczowy dowod uogolniony!

2 files changed, 57 insertions, 96 deletions

diff --git a/lic_malinka.pdf b/lic_malinka.pdf
index a0b2228..64f797a 100644
--- a/lic_malinka.pdf
+++ b/lic_malinka.pdf
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}