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Diffstat (limited to 'sections/conj_classes.tex')
| -rw-r--r-- | sections/conj_classes.tex | 47 |
1 files changed, 19 insertions, 28 deletions
diff --git a/sections/conj_classes.tex b/sections/conj_classes.tex index 9ec4b0c..d37afd5 100644 --- a/sections/conj_classes.tex +++ b/sections/conj_classes.tex @@ -90,19 +90,11 @@ automorphism group of the $\Flim(\cC)$. \end{theorem} - Before we get to the proof, let us establish some notions. If - $g\colon\Gamma\to\Gamma$ is a finite injective function, then we say that - $g$ is \emph{good} if it gives (in a natural way) an isomorphism between - $\langle \dom(g)\rangle$ and $\langle\rng(g)\rangle$, i.e. substructures - generated by $\dom(g)$ and $\rng(g)$ respectively. Of course, in our - case, $g$ is good - if and only if $[g]_{\Aut(\Gamma)} \neq\emptyset$ (because of ultrahomogeneity - of $\Gamma$). - - Also it is important to mention that an isomorphism between two finitely + Before we get to the proof, it is important to mention that an isomorphism + between two finitely generated structures is uniquely given by a map from generators of one structure to the other. This allow us to treat a finite function as an isomorphism - of finitely generated structures. + 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 @@ -115,16 +107,17 @@ By the Banach-Mazur theorem (see \ref{theorem:banach_mazur_thm}) this will prove that this class is comeagre. - Recall, $G$ has a basis consisting of open + Recall, $G$ has a basis consisting of sets $\{g\in G\mid g\upharpoonright_A = g_0\upharpoonright_A\}$ for some finite set $A\subseteq \Gamma$ and some automorphism $g_0\in G$. In other - words, a basic open set is a set of all extensions of some finite partial - automorphism $g_0$ of $\Gamma$. By $B_{g}\subseteq G$ we denote a basic - open subset given by a finite partial isomorphism $g$. Again, Note that $B_g$ + words, a basic open set is a set of all extensions of some partial + automorphism $g_0$ of finitely generated substructures of $\Gamma$. + By $B_{g}\subseteq G$ we denote a basic + open subset given by a partial isomorphism $g$. Again, Note that $B_g$ is nonempty because of ultrahomogeneity of $\Gamma$. With the use of Corollary \ref{corollary:banach-mazur-basis} we can consider - only games where both players choose finite partial isomorphisms. Namely, + only games where both players choose partial isomorphisms. Namely, player \textit{I} picks functions $f_0, f_1,\ldots$ and player \textit{II} chooses $g_0, g_1,\ldots$ such that $f_0\subseteq g_0\subseteq f_1\subseteq g_1\subseteq\ldots$, which identify @@ -144,20 +137,18 @@ For technical reasons, let $g_{-1} = \emptyset$ and $X_{-1} = \emptyset$. Enumerate the elements of the Fraïssé limit $\Gamma = \{v_0, v_1, \ldots\}$. - Suppose that player \textit{I} in the $n$-th move chooses a finite partial - automorphism $f_n$. We will construct a finite partial automorphism + Suppose that player \textit{I} in the $n$-th move chooses a partial + automorphism $f_n$. We will construct a partial automorphism $g_n\supseteq f_n$ together with a finitely generated substructure $\Gamma_n \subseteq \Gamma$ and a set $X_n\subseteq\bN^2$ such that the following properties hold: \begin{enumerate}[label=(\roman*)] - \item $g_n$ is good and - $\dom(g_n)\cup\rng(g_n)\subseteq\langle\dom(g_n)\rangle = \Gamma_n$, - i.e. $g_n$ gives an automorphism of a finitely generated - substructure $\Gamma_n$ - \item $g_n(v_n)$ and $g_n^{-1}(v_n)$ are defined, - + \item $g_n$ is is a partial automorphism of $\Gamma$ and an automorphism of + finitely generated substructure $\Gamma_n$, + \item $g_n(v_n)$ and $g_n^{-1}(v_n)$ are defined. \end{enumerate} + Before we give the third point, suppose recursively that $g_{n-1}$ already satisfy all those properties. Let us enumerate $\{\langle (A_{n,k}, \alpha_{n, k}), (B_{n,k}, \beta_{n,k}), f_{n, k}\rangle\}_{k\in\bN}$ @@ -166,14 +157,14 @@ $(A_{n,k}, \alpha_{n,k})\subseteq (B_{n,k}, \beta_{n,k})$, and $f_{n,k}$ is an embedding of $(A_{n,k}, \alpha_{n,k})$ in the $(\FrGr_{n-1}, g_{n-1})$. We allow $A_{n,k}$ to be empty. Although $g_{n-1}$ is a finite function, - we may treat it as an automorphism as we have said before. + we may treat it as a partial automorphism as we have said before. \begin{enumerate}[resume, label=(\roman*)] \item Let - $(i, j) = \min\{(\{0, 1, \ldots\} \times \bN) \setminus X_{n-1}\}$ (with the + $(i, j) = \min\{(\{0, 1, \ldots, n\} \times \bN) \setminus X_{n-1}\}$ (with the order induced by $\gamma$). Then $X_n = X_{n-1}\cup\{(i,j)\}$ and - $(B_{n,k}, \beta_{n,k})$ embeds in $(\FrGr_n, g_n)$ so that this diagram + $(B_{i,j}, \beta_{i,j})$ embeds in $(\FrGr_n, g_n)$ so that this diagram commutes: \begin{center} @@ -216,7 +207,7 @@ Take any $(B, \beta)\in\cD$. Then, there are $i, j$ such that $(B, \beta) = (B_{i, j}, \beta_{i,j})$ and $A_{i,j}=\emptyset$. By the bookkeeping there was $n$ such that - $(i, j) = \min\{\{0,1,\ldots n-1\}\times\bN\setminus X_{n-1}\}$. + $(i, j) = \min\{\{0,1,\ldots n\}\times\bN\setminus X_{n}\}$. This means that $(B, \beta)$ embeds into $(\Gamma_n, g_n)$, hence it embeds into $(\Gamma, g)$. Thus, $\cD$ is a subclass of the age of $(\Gamma, g)$. The other inclusion is obvious. Hence, the age of $(\Gamma, g)$ is $\cH$. |