From ad63d0a98d8595d3267fd81196cf8c96361bd911 Mon Sep 17 00:00:00 2001
From: Franciszek Malinka <franciszek.malinka@gmail.com>
Date: Mon, 11 Jul 2022 20:13:42 +0200
Subject: Updates updates

---
 sections/fraisse_classes.tex | 135 ++++++++++---------------------------------
 1 file changed, 31 insertions(+), 104 deletions(-)

(limited to 'sections/fraisse_classes.tex')

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}
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