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Diffstat (limited to 'sections/conj_classes.tex')
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1 files changed, 32 insertions, 10 deletions
diff --git a/sections/conj_classes.tex b/sections/conj_classes.tex index e795217..c72fd38 100644 --- a/sections/conj_classes.tex +++ b/sections/conj_classes.tex @@ -98,7 +98,7 @@ of finitely generated structures (if it yields one) and \textit{vice versa}. \begin{proof} - Let $\Gamma = \Flim(\cC)$. First, by the Theorem + Let $\Gamma = \Flim(\cC)$ and $(\Pi, \sigma) = \Flim(\cD)$. 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 $\Gamma$. @@ -245,7 +245,7 @@ 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 structure. + of the structures of $\cC$ with additional automorphism of the structure). Let $\Gamma = \Flim(\cC)$. \begin{proposition} @@ -255,13 +255,35 @@ \end{proposition} \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 $\cD$, 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 $\cD$. + Let $S = \{x\in \Gamma\mid \sigma(x) = x\}$. It is a substructure of $\Gamma$, + as it is closed under operations. For any $n$-tuple $a_1,\ldots,a_n\in S$ + and $n$-ary function symbol $f$ we have that + $f(a_1,\ldots a_n) = f(\sigma(a_1),\ldots, \sigma(a_n)) = \sigma(f(a_1,\ldots,a_n)$, + which exactly says that $f(a_1,\ldots,a_n)$ is a fixed point. Also, constant + symbols have to be fixed under $\sigma$. + + It is obvious that the + age of $S$ is $\cC$, as every structure from $\cC$ with identity embeds + into $\Gamma$, and hence into $S$. Also it is weakly ultrahomogeneous. + Take $A\subseteq B$ with embedding $f\colon A\to S$. It can be thought + as embedding of $(A, \id_A)$ into $(\Gamma,\sigma)$ and thus by its weak + ultrahomogeneity we have $\hat{f}\colon (B,\id_B)\to (\Gamma,\sigma)$ and + hence $\hat{f}$ is also and embedding of $B$ into $S$ such that the + following diagram commutes: + + \begin{center} + \begin{tikzcd} + A \ar[r, "f"] \ar[d, "\subseteq"] & S \\ + B \ar[ru, "\hat{f}"] + \end{tikzcd} + \end{center} + % First we need to show that it is an infinite substructure of $\Gamma$. + % By the theorem \ref{theorem:generic_aut_general} + % 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 $\cD$. \end{proof} \end{document} |