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Diffstat (limited to 'sections/conj_classes.tex')
| -rw-r--r-- | sections/conj_classes.tex | 14 |
1 files changed, 9 insertions, 5 deletions
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} |