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Diffstat (limited to 'sections/fraisse_classes.tex')
| -rw-r--r-- | sections/fraisse_classes.tex | 68 |
1 files changed, 50 insertions, 18 deletions
diff --git a/sections/fraisse_classes.tex b/sections/fraisse_classes.tex index 32804f2..5e45400 100644 --- a/sections/fraisse_classes.tex +++ b/sections/fraisse_classes.tex @@ -486,32 +486,64 @@ Now, take any structures $A, B\in\cC$ such that $A\subseteq B$. Without the loss of generality assume that $A = B\cap \Pi$. Let $\bar{A}\subseteq\Pi$ be the - smallest structure closed under the automorphism $\sigma$ and containing $A$. - It is finite, as $\cC$ is the age of $\Pi$. By the weak Hrushovski property, - of $\cC$ let $(\bar{B}, \beta)$ be a structure extending - $(B\cup \bar{A}, \sigma\upharpoonright_{\bar{A}})$. Again, we may assume - that $B\cup \bar{A}\subseteq \bar{B}$. Then, by the fact that $\Pi$ is a - Fraïssé limit of $\cD$ there is an embedding - $f\colon(\bar{B}, \beta)\to(\Pi, \sigma)$ - such that the following diagram commutes: + smallest substructure closed under the automorphism + $\sigma$ and containing $A$. It is finitely generated, 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 + a structure $(\bar{C}, \gamma)\in\cD$ such that + $\gamma\upharpoonright_{\bar{A}} = \sigma\upharpoonright_{\bar{A}}$. + + Then, by the weak ultrahomogeneity of $(\Pi, \sigma)$ we can find an + embedding $g$ of $(\bar{C},\gamma)$ such that the following diagram commutes: + + \begin{center} + \begin{tikzcd} + (\bar{A}, \sigma\upharpoonright_{\bar{A}}) \ar[d, "\subseteq"] \ar[r, "\subseteq"] & (\Pi, \sigma) \\ + (\bar{C}, \gamma) \ar[ur, "g"'] & + \end{tikzcd} + \end{center} - \begin{center} - \begin{tikzcd} - (A, \emptyset) \arrow[d, "\subseteq"'] \arrow[r, "\subseteq"] & (\bar{A}, \sigma\upharpoonright_A) \arrow[d, "\subseteq"] \arrow[r, "\subseteq"] & (\Pi, \sigma) \\ - (B, \sigma\upharpoonright_B) \arrow[r, dashed, "\subseteq"'] & (\bar{B}, \beta) \arrow[ur, dashed, "f"] - \end{tikzcd} - \end{center} - Then we simply get the following diagram: + Thus, we have that the following diagram commutes: \begin{center} \begin{tikzcd} - A \arrow[d, "\subseteq"'] \arrow[r, "\subseteq"] & \Pi \\ - B \arrow[ur, dashed, "f\upharpoonright_B"'] + A \ar[r, "\subseteq"] \ar[d, "\subseteq"] & \bar{A} \ar[r, "\subseteq"] \ar[d, "\subseteq"] & \Pi \\ + B \ar[r, "f"] & C \ar[r, "\subseteq"] & \bar{C} \ar[u, "g"] \\ \end{tikzcd} \end{center} - which proves that $\Pi$ is indeed a weakly ultrahomogeneous structure in $\cC$. + % + % By the weak Hrushovski property + % of $\cC$ let $(\bar{B}, \beta)$ be a structure extending + % $(B, \sigma\upharpoonright_{A})$. Again, we may assume + % that $B\cup \bar{A}\subseteq \bar{B}$. Then, by the fact that $\Pi$ is a + % Fraïssé limit of $\cD$ there is an embedding + % $f\colon(\bar{B}, \beta)\to(\Pi, \sigma)$ + % such that the following diagram commutes: + % + % + % \begin{center} + % \begin{tikzcd} + % (A, \emptyset) \arrow[d, "\subseteq"'] \arrow[r, "\subseteq"] & (\bar{A}, \sigma\upharpoonright_A) \arrow[d, "\subseteq"] \arrow[r, "\subseteq"] & (\Pi, \sigma) \\ + % (B, \sigma\upharpoonright_B) \arrow[r, dashed, "\subseteq"'] & (\bar{B}, \beta) \arrow[ur, dashed, "f"] + % \end{tikzcd} + % \end{center} + + % Then we simply get the following diagram: + % + % \begin{center} + % \begin{tikzcd} + % A \arrow[d, "\subseteq"'] \arrow[r, "\subseteq"] & \Pi \\ + % B \arrow[ur, dashed, "f\upharpoonright_B"'] + % \end{tikzcd} + % \end{center} + % + which proves that $\Pi$ is indeed a weakly ultrahomogeneous structure. Hence, it is isomorphic to $\Gamma$. \end{proof} \end{document} |