Theorem 3.23 fixed (hopefully)

3 files changed, 56 insertions, 20 deletions

diff --git a/lic_malinka.pdf b/lic_malinka.pdf
index 57df1b2..d4957dc 100644
--- a/lic_malinka.pdf
+++ b/lic_malinka.pdf
diff --git a/sections/fraisse_classes.tex b/sections/fraisse_classes.tex
index 32804f2..5e45400 100644
--- a/sections/fraisse_classes.tex
+++ b/sections/fraisse_classes.tex
@@ -486,32 +486,64 @@
 	Now, take any structures $A, B\in\cC$ such that $A\subseteq B$. Without the
 	loss of generality assume that $A = B\cap \Pi$. Let $\bar{A}\subseteq\Pi$ 
 	be the
-	smallest structure closed under the automorphism $\sigma$ and containing $A$.
-	It is finite, as $\cC$ is the age of $\Pi$. By the weak Hrushovski property,
-	of $\cC$ let $(\bar{B}, \beta)$ be a structure extending 
-	$(B\cup \bar{A}, \sigma\upharpoonright_{\bar{A}})$. Again, we may assume 
-	that $B\cup \bar{A}\subseteq \bar{B}$. Then, by the fact that $\Pi$ is a 
-	Fraïssé limit of $\cD$ there is an embedding 
-	$f\colon(\bar{B}, \beta)\to(\Pi, \sigma)$
-	such that the following diagram commutes:
+	smallest substructure closed under the automorphism 
+	$\sigma$ and containing $A$. It is finitely generated, as $\cC$ is the age 
+	of $\Pi$. 
+	Let $C$ be a finitely generated structure such that 
+	$\bar{A}\rightarrow C \leftarrow B$. Such structure exists by the JEP
+	of $\cC$. Again, we may assume without the loss of generality that
+	$\bar{A}\subseteq C$. Then $\sigma\upharpoonright_{\bar{A}}$ is a
+	partial isomorphism of $C$, hence by the WHP it can be extended to
+	a structure $(\bar{C}, \gamma)\in\cD$ such that 
+	$\gamma\upharpoonright_{\bar{A}} = \sigma\upharpoonright_{\bar{A}}$.
+
+	Then, by the weak ultrahomogeneity of $(\Pi, \sigma)$ we can find an
+	embedding $g$ of $(\bar{C},\gamma)$ such that the following diagram commutes:
+
+	\begin{center}
+	  \begin{tikzcd}
+		(\bar{A}, \sigma\upharpoonright_{\bar{A}}) \ar[d, "\subseteq"] \ar[r, "\subseteq"] & (\Pi, \sigma) \\
+		(\bar{C}, \gamma) \ar[ur, "g"'] &
+	  \end{tikzcd}
+	\end{center}
 
-    \begin{center}
-      \begin{tikzcd}
-		(A, \emptyset) \arrow[d, "\subseteq"'] \arrow[r, "\subseteq"] & (\bar{A}, \sigma\upharpoonright_A) \arrow[d, "\subseteq"] \arrow[r, "\subseteq"] & (\Pi, \sigma) \\
-		(B, \sigma\upharpoonright_B) \arrow[r, dashed, "\subseteq"'] & (\bar{B}, \beta) \arrow[ur, dashed, "f"]
-      \end{tikzcd}
-    \end{center}
 
-	Then we simply get the following diagram:
+	Thus, we have that the following diagram commutes:
 
 	\begin{center}
 	  \begin{tikzcd}
-		A \arrow[d, "\subseteq"'] \arrow[r, "\subseteq"] & \Pi \\
-		B \arrow[ur, dashed, "f\upharpoonright_B"'] 
+		A \ar[r, "\subseteq"] \ar[d, "\subseteq"] & \bar{A} \ar[r, "\subseteq"] \ar[d, "\subseteq"] & \Pi \\
+		B \ar[r, "f"] & C \ar[r, "\subseteq"] & \bar{C} \ar[u, "g"] \\
 	  \end{tikzcd}
 	\end{center}
 
-	which proves that $\Pi$ is indeed a weakly ultrahomogeneous structure in $\cC$.
+	%
+	% By the weak Hrushovski property
+	% of $\cC$ let $(\bar{B}, \beta)$ be a structure extending 
+	% $(B, \sigma\upharpoonright_{A})$. Again, we may assume 
+	% that $B\cup \bar{A}\subseteq \bar{B}$. Then, by the fact that $\Pi$ is a 
+	% Fraïssé limit of $\cD$ there is an embedding 
+	% $f\colon(\bar{B}, \beta)\to(\Pi, \sigma)$
+	% such that the following diagram commutes:
+	%
+	%
+ %    \begin{center}
+ %      \begin{tikzcd}
+	% 	(A, \emptyset) \arrow[d, "\subseteq"'] \arrow[r, "\subseteq"] & (\bar{A}, \sigma\upharpoonright_A) \arrow[d, "\subseteq"] \arrow[r, "\subseteq"] & (\Pi, \sigma) \\
+	% 	(B, \sigma\upharpoonright_B) \arrow[r, dashed, "\subseteq"'] & (\bar{B}, \beta) \arrow[ur, dashed, "f"]
+ %      \end{tikzcd}
+ %    \end{center}
+
+	% Then we simply get the following diagram:
+	%
+	% \begin{center}
+	%   \begin{tikzcd}
+	% 	A \arrow[d, "\subseteq"'] \arrow[r, "\subseteq"] & \Pi \\
+	% 	B \arrow[ur, dashed, "f\upharpoonright_B"'] 
+	%   \end{tikzcd}
+	% \end{center}
+	%
+	which proves that $\Pi$ is indeed a weakly ultrahomogeneous structure.
 	Hence, it is isomorphic to $\Gamma$.
   \end{proof}
 \end{document}
diff --git a/uwagi_29_06_22.txt b/uwagi_29_06_22.txt
index 9f6113d..0512dde 100644
--- a/uwagi_29_06_22.txt
+++ b/uwagi_29_06_22.txt
@@ -86,11 +86,11 @@ R ⊆ Π, because   is the age of Π" nie jest jasne, czemu możesz tak założy
 

 - [x] W następnym akapicie "Now, take any structure A, B": structures.

 

-- [ ] "Let ¯A be the smallest structure closed on the automorphism σ and containg A.": A tylda powinno być podstrukturą Pi i closed under, nie on.

+- [x] "Let ¯A be the smallest structure closed on the automorphism σ and containg A.": A tylda powinno być podstrukturą Pi i closed under, nie on.

 

 - [ ] "By the weak Hrushovski property, of   let (¯B,β) be a structure extending (B ∪ ¯A,σ ↾¯A).": to z grubsza działa, ale: B ∪ ¯A nie jest strukturą w C pisanym (musisz użyć JEP). A priori może się zdarzyć nawet tak, że literalnie to nie działa, bo B ∪ ¯A nie ma żadnej kompatybilnej struktury z jakichś głupich powodów, więć powinieneś to zrobićw dwóch krokach (i nie wszystkie strzałki na począku strony 12 muszą być naprawdę włożeniami). Po drugie przecinek po property jest zbędny.

 

-- [ ] "Π is indeed a weakly ultrahomogeneous structure in  " na następnej stronie jest trochę bez sensu. Wiadomo o co Ci chodzi, ale Pi nie jest w C pisanym. Po prostu Pi jest weakly ultrahomogeneous.

+- [x] "Π is indeed a weakly ultrahomogeneous structure in  " na następnej stronie jest trochę bez sensu. Wiadomo o co Ci chodzi, ale Pi nie jest w C pisanym. Po prostu Pi jest weakly ultrahomogeneous.

 

 - [ ] Ja bym przeformułował 4.1 jako remark. I napisał, że jest easy to see.

 

@@ -151,3 +151,7 @@ n and without the loss of generality we may assume that
 - [x] Dodać że wolna amalgamacja implikuje własność hrushovskiego.

 

 - [ ] Upewnić się, że używam konsekwentnie angielskiego zapisu "meagre", a nie "meager"

+

+- [ ] Poprawić definicję WHP na taką, że to chodzi o finitely generated podstruktury

+

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