Chyba poprawiony dowod 4.4

5 files changed, 90 insertions, 59 deletions

diff --git a/lic_malinka.pdf b/lic_malinka.pdf
index 03b45e9..671b753 100644
--- a/lic_malinka.pdf
+++ b/lic_malinka.pdf
diff --git a/sections/conj_classes.tex b/sections/conj_classes.tex
index 96522f5..9620220 100644
--- a/sections/conj_classes.tex
+++ b/sections/conj_classes.tex
@@ -82,7 +82,7 @@
 
   \begin{theorem}
 	\label{theorem:generic_aut_general}
-	Let $\cC$ be a Fraïssé class of finite $L$-structures. 
+	Let $\cC$ be a Fraïssé class of finitely generated $L$-structures. 
 	Let $\cD$ be the class of structures from $\cC$ with additional unary 
 	function symbol interpreted
 	as an automorphism of the structure. If $\cC$ has the weak Hrushovski property
@@ -90,11 +90,25 @@
 	automorphism group of the $\Flim(\cC)$.
   \end{theorem}
 
+  Before we get to the proof, let us establish some notions. If 
+  $g\colon\Gamma\to\Gamma$ is a finite injective function, then we say that
+  $g$ is \emph{good} if it gives (in a natural way) an isomorphism between
+  $\langle \dom(g)\rangle$ and $\langle\rng(g)\rangle$, i.e. substructures 
+  generated by $\dom(g)$ and $\rng(g)$ respectively. Of course, in our
+  case, $g$ is good
+  if and only if $[g]_{\Aut(\Gamma)} \neq\emptyset$ (becuase of ultrahomogeneity
+  of $\Gamma$.
+
+  Also it is important to mention that an isomorphism between two finitely
+  generated structures is uniquely given by a map from generators of one structure
+  to the other. This allow us to treat a finite function as an isomorphism
+  of finitely generated structure.
+
   \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
+	Let $\Gamma = \Flim(\cC)$ and $(\Pi, \sigma) = \Flim(\cD)$. First, by the Theorem
 	\ref{theorem:isomorphic_fr_lims}, we may assume without the loss of generality
-	that $\Pi = \Gamma$.
+	that $\Pi = \Gamma$. Let $G = \Aut(\Gamma)$,
+	i.e. $G$ is the automorphism group of $\Gamma$.
 	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 a subset of the $\sigma$'s conjugacy class.
@@ -105,8 +119,8 @@
 	sets $\{g\in G\mid g\upharpoonright_A = g_0\upharpoonright_A\}$ for some
 	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 $\Gamma$. By $B_{g}\subseteq G$ we denote a basic 
-	open subset given by a finite partial isomorphism $g$. Note that $B_g$
+	automorphism $g_0$ of $\Gamma$. By $B_{g}\subseteq G$ we denote a basic 
+	open subset given by a finite partial isomorphism $g$. Again, Note that $B_g$
 	is nonemty because of ultrahomogeneity of $\Gamma$.
 
 	With the use of Corollary \ref{corollary:banach-mazur-basis} we can consider 
@@ -150,20 +164,32 @@
 	For technical reasons, let $g_{-1} = \emptyset$ and
 	$X_{-1} = \emptyset$. 
 	Suppose that player \textit{I} in the $n$-th move chooses a finite partial
-	isomorphism $f_n$. We will construct a finite partial isomorphism $g_n\supseteq f_n$ 
-	and a set $X_n\subseteq\bN^2$
-	such that following properties hold:
+	automorphism $f_n$. We will construct a finite partial automorphism 
+	$g_n\supseteq f_n$ together with a finitely generated substrucutre
+	$\Gamma_n \subseteq \Gamma$ and a set $X_n\subseteq\bN^2$ 
+	such that the following properties hold:
 
 	\begin{enumerate}[label=(\roman*)]
-	  \item $g_n$ is an automorphism of the induced substructure $\Gamma_n$,
+	  \item $g_n$ is good and 
+		$\dom(g_n)\cup\rng(g_n)\subseteq\langle\dom(g_n)\rangle = \Gamma_n$,
+		i.e. $g_n$ gives an automorphism of a finitely generated 
+		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 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 substructure induced by $g_{n-1}$). Let 
+
+	  \end{enumerate}
+	  Before we give the third point, suppose recursively that $g_{n-1}$ already
+	  satisfy all those properties. Let us enumerate 
+	  $\{\langle (A_{n,k}, \alpha_{n, k}), (B_{n,k}, \beta_{n,k}), f_{n, k}\rangle\}_{k\in\bN}$
+	  all pairs of finitely generated structures with automorphisms such
+	  that the first substructure embed into the second by inclusion, 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 $(\FrGr_{n-1}, g_{n-1})$.
+	  We allow $A_{n,k}$ to be empty. Although $g_{n-1}$ is a finite function,
+	  we may treat it as an automorphism as we have said before.
+
+	  \begin{enumerate}[resume, label=(\roman*)]
+	  \item 
+		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
@@ -178,13 +204,15 @@
 		\end{center}
 	\end{enumerate}
 
-	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 $\Gamma$. The third item is the one that gives weak
-	ultrahomogeneity. Now we will show that indeed such $g_n$ may be constructed.
+	% 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 $\Gamma$. 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 $\Gamma'_n, g'_n$
-	such that $g_{n-1}\subseteq g'_n$ and $f_{i,j}$ extends to an embedding of
+	First, we will satisfy the item (iii). Namely, we will construct $\Gamma'_n, g'_n$
+	such that $g_{n-1}\subseteq g'_n$, $\Gamma_{n-1}\subseteq\Gamma'_n$,
+	$g'_n$ gives an automorphism of $\Gamma'_n$
+	and $f_{i,j}$ extends to an embedding of
 	$(B_{i,j}, \beta_{i,j})$ 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.
@@ -197,34 +225,48 @@
       \end{tikzcd}
     \end{center}
 
-	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 $\cK$ 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
+	It is important to note that $g'_n$ should be a finite function and once
+	again, as it is an automorphism of a finitely generated structure, we may
+	think it is simply a map from one generators of $\Gamma'_n$ to the 
+	others.	By the weak ultrahomogeneity of $\Gamma$, we may assume that 
+	$\Gamma'_n\subseteq \Gamma$.
+
+	% \begin{center}
+	%   \begin{tikzcd}
+	% 	B_{i,j}\cup\Gamma_{n-1} \arrow[d, "\subseteq"'] \arrow[r, "\subseteq"] & \Gamma \\
+	% 	\Gamma'_{n}\arrow[ur, dashed, "f"'] 
+	%   \end{tikzcd}
+	% \end{center}
+
+	Now, by the WHP of $\cK$ we can extend $\langle\Gamma'_n\cup\{v_n\}\rangle$ together
+	with its partial isomorphism $g'_n$ to a finitely generated structure $\Gamma_n$ 
+	together with its
+	automorphism $g_n\supseteq g'_n$ and (again by weak ultrahomogeneity)
+	without the loss of generality we
 	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 $\Gamma$ 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.
+	and is an union of an increasing chain of automorphisms of finitely generated
+	substructures.
 
-	Take any finite structure of $T$ with automorphism $(B, \beta)$. Then, there are
+	Take any $(B, \beta)\in\cD$. 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 
-	$(i, j) = \min\{\{0,1,\ldots\}\times\bN\setminus X_{n-1}\}$.
+	$(i, j) = \min\{\{0,1,\ldots n-1\}\times\bN\setminus X_{n-1}\}$.
 	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.
+	into $(\Gamma, g)$. Hence, the age of $(\Gamma, g)$ is $\cH$. 
+
+	It is also weakly ultrahomogeneous. Having $(A,\alpha)\subseteq(B,\beta)$,
+	and an embedding $f\colon(A,\alpha)\to(\Gamma,g)$, we may find $n\in\bN$
+	such that $(i,j) = \min\{\{0,1,\ldots n-1\}\times X_{n-1}\}$ and
+	$(A,\alpha) = (A_{i,j},\alpha_{i,j}), (B,\beta)=(B_{i,j},\beta_{i,j})$ and
+	$f = f_{i,j}$. This means that there is a compatbile embedding of $(B,\beta)$ into
+	$(\Gamma_n, g_n)$, which means we can also embed it into $(\Gamma, g)$.
 
+	Hence, $(\Gamma,g)\cong(\Gamma,\sigma)$.
 	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.
 	possible $g$'s we end up with) is comeagre in $G$, thus $\sigma^G$ is also
diff --git a/sections/fraisse_classes.tex b/sections/fraisse_classes.tex
index 0254280..993ca73 100644
--- a/sections/fraisse_classes.tex
+++ b/sections/fraisse_classes.tex
@@ -277,13 +277,13 @@
 	$A, B, C\in\cK$ such that $C = A\cap B$ the following diagram commutes:
 	\begin{center}
 	  \begin{tikzcd}
-		& A\sqcup B & \\
+		& A\sqcup_C B & \\
 		A \ar[ur, hook] & & B \ar[ul, hook'] \\
 		& C \ar[ur, hook] \ar[ul, hook'] & 
 	  \end{tikzcd}
 	\end{center}
 
-	$A\sqcup B$ here is an $L$-strcuture with domain $A\cup B$ such that
+	$A\sqcup_C B$ here is an $L$-strcuture with domain $A\cup B$ such that
 	for every $n$-ary symbol $R$ from $L$, $n$-tuple $\bar{a}\subseteq A\cup B$,
 	we have that $A\sqcup B\models R(\bar{a})$ if and only if [$\bar{a}\subseteq A$ and
 	$A\models R(\bar{a})$] or [$\bar{a}\subseteq B$ and $B\models R(\bar{a})$].
@@ -291,8 +291,7 @@
 
   Actually we did already implicitly worked with free amalgamation in the 
   Proposition \ref{proposition:finite-graphs-fraisse-class}, showing that
-  the class of finite strcuture is indeed a Fraïssé class.
-
+  the class of finite graphs is indeed a Fraïssé class.
 
   \subsection{Canonical amalgamation}
 
@@ -377,19 +376,9 @@
 	\end{center}
 	
 	Then, by the Fact \ref{fact:functor_iso}, $\otimes_C(\eta)$ is an automorphism
-	of the pushout diagram:
+	of the pushout diagram that looks exaclty like the diagram in the second
+	point of the Definition \ref{definition:canonical_amalgamation}.
 	
-	\begin{center}
-	  \begin{tikzcd}
-		& A\otimes_C B \ar[loop above, "\delta"] & \\
-		A \ar[ur] \ar[loop left, "\alpha"]& & B \ar[ul] \ar[loop right, "\beta"]\\
-		& C \ar[ur] \ar[ul] \ar[loop below, "\gamma"] & 
-	  \end{tikzcd}
-	\end{center}
-	
-	TODO: ten diagram nie jest do końca taki jak trzeba, trzeba w zasadzie skopiować
-	ten z definicji kanonicznej amalgamcji. Czy to nie będzie wyglądać źle?
-
 	This means that the morphism $\delta\colon A\otimes_C B\to A\otimes_C B$
 	has to be automorphism. Thus, by the fact that the diagram commutes, 
 	we have the amalgamation of $(A, \alpha)$ and $(B, \beta)$ over $(C,\gamma)$ 
diff --git a/sections/preliminaries.tex b/sections/preliminaries.tex
index 05aa3ed..2bf254c 100644
--- a/sections/preliminaries.tex
+++ b/sections/preliminaries.tex
@@ -363,7 +363,7 @@
   \end{fact}
 
   \begin{proof}
-	Suppose that $\eta_(A)$ is an isomorphism for every $A\in\cC$, where 
+	Suppose that $\eta_{A}$ is an isomorphism for every $A\in\cC$, where 
 	$\eta_{A}\colon F(A)\to G(A)$ is the morphism of the natural transformation
 	coresponding to $A$. Then $\eta^{-1}$ is simply given by the morphisms
 	$\eta^{-1}_A$.
diff --git a/uwagi_29_06_22.txt b/uwagi_29_06_22.txt
index 805fb21..0f1ef2c 100644
--- a/uwagi_29_06_22.txt
+++ b/uwagi_29_06_22.txt
@@ -144,7 +144,7 @@ n and without the loss of generality we may assume that
 

 - [x] Po 4.4 powinien być wniosek, że kanoniczna amalgamacja+whp dają konkluzję 4.4, a potem z tego, że wolna amalgamacja daje 4.4

 

-- [ ] W sekcji 4.3 brakuje założeń.

+- [x] W sekcji 4.3 brakuje założeń.

 

 - [x] Jak przekształcenie naturalne jest izomorfizmem, to ten składowe też są izomorfizmami (w dwie strony)

 

@@ -156,7 +156,7 @@ n and without the loss of generality we may assume that
 

 - [x] Dodać uwagę, że jak piszę (\Pi, \sigma) to chodzi mi o co innego niż jak piszę \Pi

 

-- [ ] "Odmętnić" początek dowódu 3.23

+- [x] "Odmętnić" początek dowódu 3.23

 

 - [x] Poprawić wielkości liter przy theorem, facts itd