Updates updates

8 files changed, 88 insertions, 131 deletions

diff --git a/lic_malinka.pdf b/lic_malinka.pdf
index d8b613e..03b45e9 100644
--- a/lic_malinka.pdf
+++ b/lic_malinka.pdf
diff --git a/lic_malinka.tex b/lic_malinka.tex
index edeb172..7e9fb07 100644
--- a/lic_malinka.tex
+++ b/lic_malinka.tex
@@ -45,8 +45,8 @@
 \DeclareMathOperator{\st}{st}

 \DeclareMathOperator{\Flim}{Flim} 

 \DeclareMathOperator{\Int}{{Int}}

-\DeclareMathOperator{\rng}{{rng}}

-\DeclareMathOperator{\dom}{{dom}}

+\DeclareMathOperator{\rng}{{Rng}}

+\DeclareMathOperator{\dom}{{Dom}}

 

 \newcommand{\cupdot}{\mathbin{\mathaccent\cdot\cup}} 

 \newcommand{\cC}{\mathcal C} 

diff --git a/sections/conj_classes.tex b/sections/conj_classes.tex
index 26229b0..96522f5 100644
--- a/sections/conj_classes.tex
+++ b/sections/conj_classes.tex
@@ -232,22 +232,25 @@
 	set $A$.
   \end{proof}
 
+  \begin{theorem}
+	\label{theorem:key-theorem}
+	Let $\cC$ be a Fraïssé class of finitely generated $L$-structures with WHP
+	and canonical amalgamation. Then $\Flim(\cC)$ has a generic automorphism.
+  \end{theorem}
+
+  \begin{proof}
+	It follows trivially from Corollary \ref{corollary:whp+canonical-iso}
+	and the above Theorem \ref{theorem:generic_aut_general}.
+  \end{proof}
+
   \subsection{Properties of the generic automorphism}
 
-  Let $\cC$ be a Fraïssé class in a finite relational language $L$ with
-  weak Hrushovski property. Let $\cH$ be the Fraïssé class of the $L$-structures
-  with additional automorphism symbol. Let $\Gamma = \Flim(\cC)$.
-
- %  \begin{proposition}
-	% Let $\sigma$ be the generic automorphism of the random graph $\FrGr$. Then
-	% the graph induced by the set of the fixed points of $\sigma$ is isomorphic
-	% to $\FrGr$.
- %  \end{proposition}
-	%
- %  \begin{proof}
-	% Let $F = \{v\in\FrGr\mid \sigma(v) = v\}$. It suffices to show that $F$ is
-	% infinite and has the random graph property.
- %  \end{proof}
+  Let $\cC$ be a Fraïssé class of finitely generated $L$-structures with
+  weak Hrushovski property and canonical amalgamation. 
+  Let $\cD$ be the Fraïssé class (by the Theorem \ref{theorem:key-theorem}
+  of the structures of $\cC$ with additional automorphism of the strucutre.
+  Let $\Gamma = \Flim(\cC)$. 
+
   \begin{proposition}
 	\label{proposition:fixed_points}
 	Let $\sigma$ be the generic automorphism of $\Gamma$. Then the set
@@ -257,11 +260,11 @@
   \begin{proof}
 	Let $S = \{x\in \Gamma\mid \sigma(x) = x\}$.
 	First we need to show that it is an infinite. By the theorem \ref{theorem:generic_aut_general}
-	we know that $(\Gamma, \sigma)$ is the Fraïssé limit of $\cH$, thus we
+	we know that $(\Gamma, \sigma)$ is the Fraïssé limit of $\cD$, thus we
 	can embed finite $L$-structures of any size with identity as an 
 	automorphism of the	structure into $(\Gamma, \sigma)$. Thus $S$ has to be
 	infinite. Also, the same argument shows that the age of the structure is
 	exactly $\cC$. It is weakly ultrahomogeneous, also by the fact that
-	$(\Gamma, \sigma)$ is in $\cH$.
+	$(\Gamma, \sigma)$ is in $\cD$.
   \end{proof}
 \end{document}
diff --git a/sections/fraisse_classes.tex b/sections/fraisse_classes.tex
index 74a8d61..0254280 100644
--- a/sections/fraisse_classes.tex
+++ b/sections/fraisse_classes.tex
@@ -236,7 +236,8 @@
 
   \begin{definition} We say that a Fraïssé class $\cK$ has the \emph{weak
     Hrushovski property} (\emph{WHP}) if for every $A\in \cK$ and an isomorphism
-	of its substructures $p\colon A\to A$ (also called a partial automorphism of $A$),
+	of its finitely generated substructures 
+	$p\colon A\to A$ (also called a partial automorphism of $A$),
 	there is some $B\in \cK$ such 
     that $p$ can be extended to an automorphism of $B$, i.e.\ there is an 
     embedding $i\colon A\to B$ and a $\bar p\in \Aut(B)$ such that the following
@@ -346,6 +347,12 @@
 	\end{itemize}
   \end{definition}
 
+  From now on in the paper, when $A$ is an $L$-strcuture and $\alpha$ is 
+  an automorphism of
+  $A$, then by $(A, \alpha)$ we mean the strucutre $A$ expanded by the
+  unary function corresping to $\alpha$, and $A$ constantly denotes the
+  $L$-strucutre.
+
   \begin{theorem}
 	\label{theorem:canonical_amalgamation_thm}
 	Let $\cK$ be a Fraïssé class of $L$-structures with canonical amalgamation. 
@@ -389,79 +396,11 @@
 	in $\cH$.
   \end{proof}
 
-  \subsection{Graphs with automorphism}
-  The language and theory of undirected graphs is fairly simple. We extend the
-  language by one unary symbol $\sigma$ and interpret it as an arbitrary
-  automorphism on the graph structure. It turns out that the class of such
-  structures is a Fraïssé class.
-
-  \begin{proposition}
-    Let $\cH$ be the class of all finite graphs with an automorphism, i.e.
-    structures in the language $(E, \sigma)$ such that $E$ is a symmetric
-    relation and $\sigma$ is an automorphism on the structure. $\cH$ is
-    a Fraïssé class.
-  \end{proposition}
-  \begin{proof}
-    Countability and HP are obvious, JEP follows by the same argument as in
-    plain graphs. We need to show that the class has the amalgamation property.
-
-    Take any $(A, \alpha), (B, \beta), (C,\gamma)\in\cH$ such that $A$ embeds
-    into $B$ and $C$. Without the loss of generality we may assume that 
-	$B\cap C = A$ and $\alpha\subseteq\beta,\gamma$. 
-	Let $D$ be the amalgamation of $B$ and $C$ over $A$ as in
-    the proof for the plain graphs. We will define the automorphism
-    $\delta\in\Aut(D)$ so it extends $\beta$ and $\gamma$. 
-	We let $\delta\upharpoonright_B = \beta, \delta\upharpoonright_C = \gamma$.
-
-	Let us check that the definition is correct. We have to show that 
-	$(uE_Dv\leftrightarrow \delta(u)E_D\delta(v))$ holds for any $u, v\in
-	D$. We have two cases:
-    \begin{itemize}
-      \item $u, v\in X$, where $X$ is either $B$ or $C$. This case is trivial.
-      \item $u\in B\setminus A, v\in C\setminus A$. Then
-        $\delta(u)=\beta(u)\in B\setminus A$, similarly $\delta(v)=\gamma(v)\in
-        C\setminus A$. This follows from the fact that $\beta\upharpoonright_A
-        = \alpha$, so for any $w\in A$ it holds that 
-		$\beta^{-1}(w)=\alpha^{-1}(w)\in A$,
-        similarly for $\gamma$. Thus, from the construction of $D$, $\neg uE_Dv$
-        and $\neg \delta(u)E_D\delta(v)$.
-    \end{itemize}
-  \end{proof}
-
-  The proposition says that there is a Fraïssé limit for the class $\cH$ of finite
-  graphs with automorphisms. We shall call it $(\FrAut, \sigma)$. Not
-  surprisingly, $\FrAut$ is in fact a random graph.
-
-  \begin{proposition}
-	\label{proposition:graph-aut-is-normal}
-	The Fraïssé limit of $\cH$ interpreted as a plain graph (i.e. as a reduct
-	to the language of graphs) is isomorphic to the random graph $\FrGr$. 
-  \end{proposition}
-
-  \begin{proof}
-    It is enough to show that $\FrAut = \Flim(\cH)$ has the random graph
-    property. Take any finite disjoint $X, Y\subseteq \FrAut$. Without the loss
-    of generality assume that $X\cup Y$ is $\sigma$-invariant, i.e.
-    $\sigma(v)\in X\cup Y$ for $v\in X\cup Y$. This assumption can be made 
-    because there are no infinite orbits in $\sigma$, which in turn is due to
-    the fact that $\cH$ is the age of $\FrAut$. 
-
-    Let $G_{XY}$ be the graph induced by $X\cup Y$. Take $H=G_{XY}\sqcup {v}$ 
-    as a supergraph of $G_{XY}$ with one new vertex $v$ connected to all 
-    vertices of $X$ and to none of $Y$. By the proposition
-    \ref{proposition:finite-graphs-whp} we can extend $H$ together with its
-    partial isomorphism $\sigma\upharpoonright_{X\cup Y}$ to a graph $R$ with
-    automorphism $\tau$. Once again, without the loss of generality we can
-    assume that $R\subseteq\FrAut$, because $\cH$ is the age of $\FrAut$. But
-    $R\upharpoonright_{G_{XY}}$ together with $\tau\upharpoonright_{G_{XY}}$
-    are isomorphic to the $G_{XY}$ with $\sigma\upharpoonright_{G_{XY}}$.
-
-    Thus, by ultrahomogeneity of $\FrAut$ this isomorphism extends to an 
-    automorphism $\theta$ of $(\FrAut, \sigma)$. Then $\theta(v)$ is the vertex
-    that is connected to all vertices of $X$ and none of $Y$, because
-    $\theta[R\upharpoonright_X] = X, \theta[R\upharpoonright_Y] = Y$.
-  \end{proof}
-
+  The following theorem is one of the most important in construction of
+  the generic automorphism given in the next section. Together with canonical
+  amalgamation it gives a general fact about Fraïssé classes, namely it says
+  that expanding a Fraïssé class with an automorphism of the structures
+  does not change the limit.
 
   \begin{theorem}
 	\label{theorem:isomorphic_fr_lims}
@@ -483,11 +422,11 @@
 	$(\Pi, \sigma)$, then obviously $B\in\cC$ by the definition of $\cD$.
 	Hence, $\cC$ is indeed the age of $\Pi$.
 
-	Now, take any structures $A, B\in\cC$ such that $A\subseteq B$. We will
-	find an embedding of $B$ into $\Pi$ to show that $\Pi$ is indeed weakly
-	homogeneous.
+	Now, to show that $\Pi$ is weakly homogeneous, take any structures $A, B\in\cC$ 
+	such that $A\subseteq B$ with a fixed embedding of $A$ into $\Pi$. 
 	Without the
-	loss of generality assume that $A = B\cap \Pi$. Let $\bar{A}\subseteq\Pi$ 
+	loss of generality assume that $A = B\cap \Pi$ (i.e. $A$ embeds into $\Pi$
+	by inclusion). Let $\bar{A}\subseteq\Pi$ 
 	be the
 	smallest substructure closed under the automorphism 
 	$\sigma$ and containing $A$. It is finitely generated as an $L$-structure, 
@@ -520,33 +459,21 @@
 	  \end{tikzcd}
 	\end{center}
 
-	%
-	% 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}
+
+  \begin{corollary}
+	\label{corollary:whp+canonical-iso}
+	Let $\cC$ be a Fraïssé class of finitely generated $L$-structures
+	with WHP and canonical amalgamation. Let
+	$\cD$ be the class consisting of structures from $\cC$ with an additional
+	automorphism. Let $\Gamma = \Flim(\cC)$ and $\Pi = \Flim(\cD)$.
+	Then $\Gamma \cong \Pi\mid_L$.
+  \end{corollary}
+
+  \begin{proof}
+	It follows from Theorems \ref{theorem:canonical_amalgamation_thm} and
+	\ref{theorem:isomorphic_fr_lims}.
+  \end{proof}
 \end{document}
diff --git a/sections/introduction-pl.tex b/sections/introduction-pl.tex
index 02b6cbe..bd6e2f1 100644
--- a/sections/introduction-pl.tex
+++ b/sections/introduction-pl.tex
@@ -27,7 +27,7 @@
   \ref{definition:generic_automorphism}, co zostało po raz pierwsze zdefiniowane
   i udowodnione przez Trussa w \cite{truss_gen_aut}.
 
-  Kluczowe twierdzenie \ref{theorem:generic_aut_general} mówi, że klasa
+  Kluczowe twierdzenie \ref{theorem:key-theorem} mówi, że klasa
   Fraïsségo z kanoniczną amalgamacją i słabą własnością Hrushovskiego
   ma generyczny automorfizm. Istnienie takiego automorfizmu w tym przypadku
   wynika z wcześniejszych klasycznych wyników Ivanova \cite{ivanov_1999}
diff --git a/sections/introduction.tex b/sections/introduction.tex
index 34b42f8..655cba7 100644
--- a/sections/introduction.tex
+++ b/sections/introduction.tex
@@ -26,7 +26,7 @@
   \ref{definition:generic_automorphism}, which was first proved
   by Truss in \cite{truss_gen_aut}, where he also introduced the term. 
 
-  The key Theorem \ref{theorem:generic_aut_general}
+  The key Theorem \ref{theorem:key-theorem}
   says that a Fraïssé class with canonical amalgamation and weak Hrushovski
   property has a generic automorphism. The fact that such an automorphism
   exists in this case follows from the classical results of Ivanov \cite{ivanov_1999}
diff --git a/sections/preliminaries.tex b/sections/preliminaries.tex
index 82e64b4..05aa3ed 100644
--- a/sections/preliminaries.tex
+++ b/sections/preliminaries.tex
@@ -306,7 +306,11 @@
   $\Obj(\cC)$, but most often simply as $\cC$) and collection of \emph{morphisms}
   $\Mor(A, B)$ between each pair of objects $A, B\in \cC$. We require that
   for each pair of morphisms $f\colon B\to C$, $g\colon A\to B$ there is a 
-  morphism $f\circ g\colon A\to C$. For every $A\in\cC$ there is an 
+  morphism $f\circ g\colon A\to C$. If $f\colon A\to B$ then we say
+  that $A$ is the domain of $f$ ($\dom{f}$) and that $B$ is the range of
+  $f$ ($\rng{f}$).
+
+  For every $A\in\cC$ there is an 
   \emph{identity morphism} $\id_A$ such that for any morphism $f\in \Mor(A, B)$
   we have that $f\circ id_A = \id_B \circ f$.
 
@@ -349,6 +353,29 @@
     \end{center}
   \end{definition}
 
+  Natural transformation has, \textit{nomen omen}, natural properties. One
+  particularly interesting to us is the following fact.
+
+  \begin{fact}
+	Let $\eta$ be a natural transformation of functors $F, G$ from category
+	$\cC$ to $\cD$. Then $\eta$ is an isomorphism if and only if
+	all of the component morphisms are isomorphisms.
+  \end{fact}
+
+  \begin{proof}
+	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
+	$\eta^{-1}_A$.
+
+	Now assume that $\eta$ is an isomorphism, i.e. $\eta^{-1}\circ\eta = \id_F$.
+	\textit{Ad contrario} assume that there is $A\in\cC$ such that the component
+	morphism $\eta_A\colon F(A)\to G(A)$ is not an isomorphism. It means
+	that $\eta_A^{-1}\circ\eta_A \neq id_A$, hence 
+	$F(A) = \dom(\eta^{-1}\circ\eta)(A) \neq \rng(\eta^{-1}\circ\eta)(A) = F(A)$,
+	which is obviously a contradiction.
+  \end{proof}
+
   \begin{definition}
 	In category theory, a \emph{diagram} of type $\mathcal{J}$ in category $\cC$
 	is a functor $D\colon \mathcal{J}\to\cC$. $\mathcal{J}$ is called the
diff --git a/uwagi_29_06_22.txt b/uwagi_29_06_22.txt
index 33d8715..805fb21 100644
--- a/uwagi_29_06_22.txt
+++ b/uwagi_29_06_22.txt
@@ -142,23 +142,23 @@ n and without the loss of generality we may assume that
 

 - [ ] "With a similar argument we can see that (Γ, g) is weakly ultrahomogeneous." to trochę oszustwo, wytłumacz to nieco dokładniej.

 

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

+- [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.

 

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

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

 

 - [x] Dodać że wolna amalgamacja implikuje własność hrushovskiego.

 

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

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

 

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

+- [x] 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

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

 

 - [ ] "Odmętnić" początek dowódu 3.23

 

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

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

 

 - [ ] Przykłady (porządki liniowe, porządki cykliczne, przestrzenie liniowe)