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authorFranciszek Malinka <franciszek.malinka@gmail.com>2022-08-24 23:22:12 +0200
committerFranciszek Malinka <franciszek.malinka@gmail.com>2022-08-24 23:22:12 +0200
commit37bb95dea0ac896d9c409fe64f3c1ce99ca43bb2 (patch)
treea6ff9e51b50fe9d6c47edddd4539846833788f76
parenta15a0040023eb4f8f2b9d9653789063b86ccbe62 (diff)
Fixes
-rw-r--r--lic_malinka.pdfbin494347 -> 494752 bytes
-rw-r--r--sections/conj_classes.tex14
-rw-r--r--sections/examples.tex63
3 files changed, 44 insertions, 33 deletions
diff --git a/lic_malinka.pdf b/lic_malinka.pdf
index cb7e801..ebe27fa 100644
--- a/lic_malinka.pdf
+++ b/lic_malinka.pdf
Binary files differ
diff --git a/sections/conj_classes.tex b/sections/conj_classes.tex
index 87fa416..e795217 100644
--- a/sections/conj_classes.tex
+++ b/sections/conj_classes.tex
@@ -85,9 +85,10 @@
Let $\cC$ be a Fraïssé class of finitely generated $L$-structures.
Let $\cD$ be the class of structures from $\cC$ with additional unary
function symbol interpreted
- as an automorphism of the structure. If $\cC$ has the weak Hrushovski property
- and $\cD$ is a Fraïssé class, then there is a comeagre conjugacy class in the
- automorphism group of the $\Flim(\cC)$.
+ as an automorphism of the structure. If $\cC$ has the weak Hrushovski
+ property, $\cD$ is a Fraïssé class and $(\Pi, \sigma) = \Flim(\cD)$, then
+ $\Aut(\cC)$ has a comeagre conjugacy class and $\sigma$ is in this class
+ (i.e. $\sigma$ is a generic automorphism).
\end{theorem}
Before we get to the proof, it is important to mention that an isomorphism
@@ -97,7 +98,7 @@
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
+ Let $\Gamma = \Flim(\cC)$. 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 $\Gamma$.
@@ -228,7 +229,10 @@
\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.
+ and canonical amalgamation. Let $\cD$ be the class of structures from $\cC$ with
+ additional unary function symbol interpreted as an automorphism of the structure.
+ Then $\cD$ is a Fraïssé class and for $(\Pi, \sigma) = \Flim(\cD)$
+ we have that the conjugacy class of $\sigma$ is comeagre in $\Aut(\cC)$.
\end{theorem}
\begin{proof}
diff --git a/sections/examples.tex b/sections/examples.tex
index fdb8c6a..3d276ff 100644
--- a/sections/examples.tex
+++ b/sections/examples.tex
@@ -1,6 +1,41 @@
\documentclass[../lic_malinka.tex]{subfiles}
\begin{document}
+ In this section we give examples and anti-examples of Fraïssé classes
+ with WHP or CAP.
+
+
+ \begin{example}
+ The class of all finite graphs $\cG$ is a Fraïssé class with WHP and free
+ amalgamation.
+ \end{example}
+
+ We have already shown this fact. Thus get that the random graph has a generic
+ automorphism.
+
+ \begin{example}
+ A $K_n$-free graph is a graph with no $n$-clique as its subgraph.
+ Let $\cG_n$ be the class of finite \emph{$K_n$-free} graphs. $\cG_n$
+ is a Fraïssé class with WHP and free amalgamation.
+ \end{example}
+
+ Showing that $\cG_n$ is indeed a Fraïssé class is almost the same as in
+ normal graphs, together with free amalgamation. WHP is trickier and the proof
+ can be seen in \cite{eppa_presentation} Theorem 3.6. Hence, $\Flim(\cG_n)$
+ has a generic automorphism.
+
+ \begin{example}
+ The class $\cV$ of all finitely generated vector spaces over a countable field
+ is a Fraïssé class with WHP and CAP.
+ \end{example}
+
+ Vector spaces of the same dimension are isomorphic, thus it is obvious that
+ $\cV$ is essentially countable. Also $HP$ and $JEP$ are obvious, as we can
+ always embed space with smaller dimension into the bigger one. Amalgamation
+ works exactly the same. In fact, such amalgamation is indeed canonical.
+
+ Now we give some anti-examples:
+
\begin{example}
Let $\cL$ be the class of all finite linear orderings. Then:
\begin{enumerate}
@@ -155,32 +190,4 @@
Thus there cannot be a dense conjugacy class in $\Aut(\Sigma)$ and so there's
no generic automorphism.
- \begin{example}
- The class $\cV$ of all finitely generated vector spaces over a countable field
- is a Fraïssé class with WHP and CAP.
- \end{example}
-
- Vector spaces of the same dimension are isomorphic, thus it is obvious that
- $\cV$ is essentially countable. Also $HP$ and $JEP$ are obvious, as we can
- always embed space with smaller dimension into the bigger one. Amalgamation
- works exactly the same. In fact, such amalgamation is indeed canonical.
-
- \begin{example}
- The class of all finite graphs $\cG$ is a Fraïssé class with WHP and free
- amalgamation.
- \end{example}
-
- We have already shown this fact. Thus get that the random graph has a generic
- automorphism.
-
- \begin{example}
- A $K_n$-free graph is a graph with no $n$-clique as its subgraph.
- Let $\cG_n$ be the class of finite \emph{$K_n$-free} graphs. $\cG_n$
- is a Fraïssé class with WHP and free amalgamation.
- \end{example}
-
- Showing that $\cG_n$ is indeed a Fraïssé class is almost the same as in
- normal graphs, together with free amalgamation. WHP is trickier and the proof
- can be seen in \cite{eppa_presentation} Theorem 3.6. Hence, $\Flim(\cG_n)$
- has a generic automorphism.
\end{document}