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Diffstat (limited to 'sections/fraisse_classes.tex')
| -rw-r--r-- | sections/fraisse_classes.tex | 11 |
1 files changed, 7 insertions, 4 deletions
diff --git a/sections/fraisse_classes.tex b/sections/fraisse_classes.tex index 5e45400..5f3d833 100644 --- a/sections/fraisse_classes.tex +++ b/sections/fraisse_classes.tex @@ -483,17 +483,20 @@ $(\Pi, \sigma)$, 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 $A\subseteq B$. Without the + 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. + Without the loss of generality assume that $A = B\cap \Pi$. Let $\bar{A}\subseteq\Pi$ be the smallest substructure closed under the automorphism - $\sigma$ and containing $A$. It is finitely generated, as $\cC$ is the age - of $\Pi$. + $\sigma$ and containing $A$. It is finitely generated as an $L$-structure, + as $\cC$ is the age of $\Pi$. Let $C$ be a finitely generated structure such that $\bar{A}\rightarrow C \leftarrow B$. Such structure exists by the JEP of $\cC$. Again, we may assume without the loss of generality that $\bar{A}\subseteq C$. Then $\sigma\upharpoonright_{\bar{A}}$ is a - partial isomorphism of $C$, hence by the WHP it can be extended to + partial automorphism of $C$, hence by the WHP it can be extended to a structure $(\bar{C}, \gamma)\in\cD$ such that $\gamma\upharpoonright_{\bar{A}} = \sigma\upharpoonright_{\bar{A}}$. |