diff options
| -rw-r--r-- | lic_malinka.pdf | bin | 494347 -> 494752 bytes | |||
| -rw-r--r-- | sections/conj_classes.tex | 14 | ||||
| -rw-r--r-- | sections/examples.tex | 63 |
3 files changed, 44 insertions, 33 deletions
diff --git a/lic_malinka.pdf b/lic_malinka.pdf Binary files differindex cb7e801..ebe27fa 100644 --- a/lic_malinka.pdf +++ b/lic_malinka.pdf 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} |