Oby ostatnie poprawki

7 files changed, 32 insertions, 72 deletions

diff --git a/lic_malinka.pdf b/lic_malinka.pdf
index 0529e44..6354005 100644
--- a/lic_malinka.pdf
+++ b/lic_malinka.pdf
diff --git a/licmalinka.bib b/licmalinka.bib
index 6c1237c..26dd13f 100644
--- a/licmalinka.bib
+++ b/licmalinka.bib
@@ -53,13 +53,14 @@
 	year = {2007}
 },
 @article{extending_iso_graphs,
-	title={http://math.univ-lyon1.fr/~milliet/grapheanglais.pdf},
+	title={Extending partial isomorphisms of finite graphs},
     url={http://math.univ-lyon1.fr/~milliet/grapheanglais.pdf},
 	author={Cédric Milliet},
 	year={2004}
 },
 @article{hrushovski_extending_iso,
   author={Ehud Hrushovski},
+  title={Extending partial isomorphisms of graphs},
   year={1992},
   journal={Combinatorica},
   volume={12},
@@ -68,13 +69,6 @@
 },
 @book{siniora2017automorphism,
   title={Automorphism Groups of Homogeneous Structures},
-  author={Siniora, D.N.},
-  url={https://books.google.pl/books?id=-qiZtAEACAAJ},
-  year={2017},
-  publisher={University of Leeds (Department of Pure Mathematics)}
-}
-@book{siniora2017automorphism,
-  title={Automorphism Groups of Homogeneous Structures},
   author={Daoud Nasri Siniora},
   url={https://core.ac.uk/download/pdf/83934818.pdf},
   year={2017},
diff --git a/sections/conj_classes.tex b/sections/conj_classes.tex
index c0120d3..9ec4b0c 100644
--- a/sections/conj_classes.tex
+++ b/sections/conj_classes.tex
@@ -46,7 +46,7 @@
 	$\Aut(M)$: take any finite $\gamma\colon M\to M$ such that $[\gamma]_{\Aut(M)}$
 	is nonempty. Then also 
 	$\bigcup_{\beta\in B_{n,k}} [\beta]_{\Aut(M)}\cap[\gamma]_{\Aut(M)}\neq\emptyset$,
-	one can easily construct a permutation that extends $\gamma$ and have at least
+	one can easily construct a permutation that extends $\gamma$ and has at least
 	$k$ many $n$-cycles.
 
 	Now we see that $A = \bigcap_{n=1}^{\infty} A_n$ is a comeagre set consisting
@@ -97,21 +97,21 @@
   generated by $\dom(g)$ and $\rng(g)$ respectively. Of course, in our
   case, $g$ is good
   if and only if $[g]_{\Aut(\Gamma)} \neq\emptyset$ (because of ultrahomogeneity
-  of $\Gamma$.
+  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.
+  of finitely generated structures.
 
   \begin{proof}
 	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 $\Pi = \Gamma$. 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.
+	We will construct a winning strategy for the second player in the Banach-Mazur game
+	(see \ref{definition:banach-mazur-game})
+	on the topological space $G$ with $A$ being $\sigma$'s conjugacy class. 
 	By the Banach-Mazur theorem (see \ref{theorem:banach_mazur_thm}) this will prove 
 	that this class is comeagre.
 
@@ -129,15 +129,6 @@
 	chooses $g_0, g_1,\ldots$ such that
 	$f_0\subseteq g_0\subseteq f_1\subseteq g_1\subseteq\ldots$, which identify
 	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 Fraïssé limit 
-	% $\Gamma$ such that $(\Gamma, g) = \Flim{\cD}$. 
-	% Precisely, we will find $g_i$'s such that 
-	% $\bigcap^{\infty}_{i=0}B_{g_i} = \{g\}$ and 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 $G$.
 	
 	Our goal is to choose $g_i$ in such a manner that 
 	$\bigcap^{\infty}_{i=0}B_{g_i} = \{g\}$ and $(\Gamma, g)$ is ultrahomogeneous
@@ -145,24 +136,14 @@
 	that $(\Gamma, \sigma)\cong (\Gamma, g)$, thus by the Fact \ref{fact:conjugacy}
 	we have that $\sigma$ and $g$ conjugate.
 
-	% Once again, by the Fraïssé theorem and by Lemma 
-	% \ref{lemma:weak_ultrahom} constructing $g_i$'s in a way such that
-	% 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. 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
 	a well ordering on $\bN\times\bN$. This will prove useful later, as the
 	main ingredient of the proof will be a bookkeeping argument.
 
 	For technical reasons, let $g_{-1} = \emptyset$ and
-	$X_{-1} = \emptyset$. 
+	$X_{-1} = \emptyset$. Enumerate the elements of the Fraïssé limit 
+	$\Gamma = \{v_0, v_1, \ldots\}$.
 	Suppose that player \textit{I} in the $n$-th move chooses a finite partial
 	automorphism $f_n$. We will construct a finite partial automorphism 
 	$g_n\supseteq f_n$ together with a finitely generated substructure
@@ -204,26 +185,13 @@
 		\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, 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.
+	done by the fact, that $\cD$ has the amalgamation property. 
 
-    \begin{center}
-      \begin{tikzcd}
-                                  & (\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}
 
 	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
@@ -231,14 +199,7 @@
 	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
+	Now, by the WHP of $\cC$ 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)
@@ -247,7 +208,7 @@
 	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.
+	of $\Gamma$ such that $(\Gamma, g)$ has the age $\cD$ and is weakly ultrahomogeneous.
 	It is of course an automorphism of $\Gamma$ as it is defined for every $v\in\Gamma$
 	and is an union of an increasing chain of automorphisms of finitely generated
 	substructures.
@@ -257,7 +218,8 @@
 	By the bookkeeping there was $n$ such that 
 	$(i, j) = \min\{\{0,1,\ldots n-1\}\times\bN\setminus X_{n-1}\}$.
 	This means that $(B, \beta)$ embeds into $(\Gamma_n, g_n)$, hence it embeds
-	into $(\Gamma, g)$. Hence, the age of $(\Gamma, g)$ is $\cH$. 
+	into $(\Gamma, g)$. Thus, $\cD$ is a subclass of the age of $(\Gamma, g)$.
+	The other inclusion is obvious.	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 $n\in\bN$
@@ -267,11 +229,9 @@
 	$(\Gamma_n, g_n)$, 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
-	comeagre and $\sigma$ is a generic automorphism, as it contains a comeagre
-	set $A$.
+	By this we know that $g$ and $\sigma$ are conjugate in $G$, thus player \textit{II}
+	have a winning strategy in the Banach-Mazur game with $A=\sigma^G$,
+	thus $\sigma^G$ is comeagre in $G$ and $\sigma$ is a generic automorphism.
   \end{proof}
 
   \begin{theorem}
diff --git a/sections/fraisse_classes.tex b/sections/fraisse_classes.tex
index 87647c6..1126dee 100644
--- a/sections/fraisse_classes.tex
+++ b/sections/fraisse_classes.tex
@@ -283,13 +283,14 @@
 	  \end{tikzcd}
 	\end{center}
 
+	and $A\sqcup_C B\in\cC$.
 	$A\sqcup_C B$ here is an $L$-structure 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
+	we have that $A\sqcup_C B\models R(\bar{a})$ if and only if [$\bar{a}\subseteq A$ and
 	$A\models R(\bar{a})$] or [$\bar{a}\subseteq B$ and $B\models R(\bar{a})$].
   \end{definition}
 
-  Actually we did already implicitly worked with free amalgamation in the 
+  Actually we did already implicitly work with free amalgamation in the 
   Proposition \ref{proposition:finite-graphs-fraisse-class}, showing that
   the class of finite graphs is indeed a Fraïssé class.
 
@@ -298,7 +299,7 @@
   Recall, $\Cospan(\cC)$, $\Pushout(\cC)$ are the categories of cospan
   and pushout diagrams of the category $\cC$. We have also denoted the notion
   of cospans and pushouts with a fixed base structure $C$ denoted
-  as $\Cospan_C(\cC)$ and $Pushout_C(\cC)$.
+  as $\Cospan_C(\cC)$ and $\Pushout_C(\cC)$.
 
   \begin{definition}
 	\label{definition:canonical_amalgamation}
@@ -346,6 +347,10 @@
 	\end{itemize}
   \end{definition}
 
+  \begin{remark}
+	Free amalgamation is canonical.
+  \end{remark}
+
   From now on in the paper, when $A$ is an $L$-structure and $\alpha$ is 
   an automorphism of
   $A$, then by $(A, \alpha)$ we mean the structure $A$ expanded by the
diff --git a/sections/introduction-pl.tex b/sections/introduction-pl.tex
index bd6e2f1..1b06733 100644
--- a/sections/introduction-pl.tex
+++ b/sections/introduction-pl.tex
@@ -38,6 +38,6 @@
   grami Banacha-Mazura, które są dobrze znanym narzędziem w deskryptywnej
   teorii mnogości.
 
-  Opisana konstrukcja generycznego automorfizmu okazuje się pomocna w dowodzeniu
-  niektórych własności tego automorfizmu (patrz \ref{proposition:fixed_points}).
+  % Opisana konstrukcja generycznego automorfizmu okazuje się pomocna w dowodzeniu
+  % niektórych własności tego automorfizmu (patrz \ref{proposition:fixed_points}).
 \end{document}
diff --git a/sections/introduction.tex b/sections/introduction.tex
index 655cba7..6cb432e 100644
--- a/sections/introduction.tex
+++ b/sections/introduction.tex
@@ -36,7 +36,7 @@
   this by using the Banach-Mazur games, a well known method in the descriptive
   set theory, which proves useful in the study of comeagre sets.
 
-  Finally, we show how this construction of the generic automorphism can be
-  used to deduce some properties of generic automorphisms 
-  (see \ref{proposition:fixed_points}, (COŚ JESZCE)).
+  % Finally, we show how this construction of the generic automorphism can be
+  % used to deduce some properties of generic automorphisms 
+  % (see \ref{proposition:fixed_points}, (COŚ JESZCE)).
 \end{document}
diff --git a/sections/preliminaries.tex b/sections/preliminaries.tex
index 0a1b202..b27cd69 100644
--- a/sections/preliminaries.tex
+++ b/sections/preliminaries.tex
@@ -76,6 +76,7 @@
   \end{definition}
 
   \begin{definition} 
+	\label{definition:banach-mazur-game}
 	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