From ef3fc25228cc13c9f39192813653b1b6dc339406 Mon Sep 17 00:00:00 2001
From: Franciszek Malinka <franciszek.malinka@gmail.com>
Date: Sun, 17 Jul 2022 23:22:37 +0200
Subject: Szkola orlow finish!

---
 sections/conj_classes.tex | 47 +++++++++++++++++++----------------------------
 1 file changed, 19 insertions(+), 28 deletions(-)

(limited to 'sections/conj_classes.tex')

diff --git a/sections/conj_classes.tex b/sections/conj_classes.tex
index 9ec4b0c..d37afd5 100644
--- a/sections/conj_classes.tex
+++ b/sections/conj_classes.tex
@@ -90,19 +90,11 @@
 	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 $[g]_{\Aut(\Gamma)} \neq\emptyset$ (because of ultrahomogeneity
-  of $\Gamma$).
-
-  Also it is important to mention that an isomorphism between two finitely
+  Before we get to the proof, 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 structures.
+  of finitely generated structures (if it yields one) and \textit{vice versa}. 
 
   \begin{proof}
 	Let $\Gamma = \Flim(\cC)$ and $(\Pi, \sigma) = \Flim(\cD)$. First, by the Theorem
@@ -115,16 +107,17 @@
 	By the Banach-Mazur theorem (see \ref{theorem:banach_mazur_thm}) this will prove 
 	that this class is comeagre.
 
-	Recall, $G$ has a basis consisting of open
+	Recall, $G$ has a basis consisting of 
 	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
-	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 $B_g$
+	words, a basic open set is a set of all extensions of some partial
+	automorphism $g_0$ of finitely generated substructures of $\Gamma$. 
+	By $B_{g}\subseteq G$ we denote a basic 
+	open subset given by a partial isomorphism $g$. Again, Note that $B_g$
 	is nonempty because of ultrahomogeneity of $\Gamma$.
 
 	With the use of Corollary \ref{corollary:banach-mazur-basis} we can consider 
-	only games where both players choose finite partial isomorphisms. Namely,
+	only games where both players choose partial isomorphisms. Namely,
 	player \textit{I} picks functions $f_0, f_1,\ldots$ and player \textit{II}
 	chooses $g_0, g_1,\ldots$ such that
 	$f_0\subseteq g_0\subseteq f_1\subseteq g_1\subseteq\ldots$, which identify
@@ -144,20 +137,18 @@
 	For technical reasons, let $g_{-1} = \emptyset$ and
 	$X_{-1} = \emptyset$. 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 
+	Suppose that player \textit{I} in the $n$-th move chooses a partial
+	automorphism $f_n$. We will construct a partial automorphism 
 	$g_n\supseteq f_n$ together with a finitely generated substructure
 	$\Gamma_n \subseteq \Gamma$ and a set $X_n\subseteq\bN^2$ 
 	such that the following properties hold:
 
 	\begin{enumerate}[label=(\roman*)]
-	  \item $g_n$ is 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 $g_n$ is is a partial automorphism of $\Gamma$ and an automorphism of
+		finitely generated substructure $\Gamma_n$,
+	  \item $g_n(v_n)$ and $g_n^{-1}(v_n)$ are defined.
 	  \end{enumerate}
+
 	  Before we give the third point, suppose recursively that $g_{n-1}$ already
 	  satisfy all those properties. Let us enumerate 
 	  $\{\langle (A_{n,k}, \alpha_{n, k}), (B_{n,k}, \beta_{n,k}), f_{n, k}\rangle\}_{k\in\bN}$
@@ -166,14 +157,14 @@
 	  $(A_{n,k}, \alpha_{n,k})\subseteq (B_{n,k}, \beta_{n,k})$, 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.
+	  we may treat it as a partial 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
+		$(i, j) = \min\{(\{0, 1, \ldots, n\} \times \bN) \setminus X_{n-1}\}$ (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
+		$(B_{i,j}, \beta_{i,j})$ embeds in $(\FrGr_n, g_n)$ so that this diagram
 		commutes:
 
 		\begin{center}
@@ -216,7 +207,7 @@
 	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 
-	$(i, j) = \min\{\{0,1,\ldots n-1\}\times\bN\setminus X_{n-1}\}$.
+	$(i, j) = \min\{\{0,1,\ldots n\}\times\bN\setminus X_{n}\}$.
 	This means that $(B, \beta)$ embeds into $(\Gamma_n, g_n)$, hence it embeds
 	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 $\cH$. 
-- 
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