From e0fea5a2063f6babc496f593dbb05f8814bddc67 Mon Sep 17 00:00:00 2001 From: Franciszek Malinka Date: Sun, 8 May 2022 19:42:41 +0200 Subject: More general proposition 3.18 --- lic_malinka.pdf | Bin 391756 -> 399632 bytes lic_malinka.tex | 85 ++++++++++++++++++++++++++++++++++++++++++++++++++++++-- 2 files changed, 83 insertions(+), 2 deletions(-) diff --git a/lic_malinka.pdf b/lic_malinka.pdf index edf6b66..a0b2228 100644 Binary files a/lic_malinka.pdf and b/lic_malinka.pdf differ diff --git a/lic_malinka.tex b/lic_malinka.tex index 5e52034..9b218ee 100644 --- a/lic_malinka.tex +++ b/lic_malinka.tex @@ -47,6 +47,7 @@ \newcommand{\cupdot}{\mathbin{\mathaccent\cdot\cup}} \newcommand{\cC}{\mathcal C} +\newcommand{\cD}{\mathcal D} \newcommand{\cV}{\mathcal{V}} \newcommand{\cU}{\mathcal{U}} \newcommand{\cG}{\mathcal{G}} @@ -144,7 +145,8 @@ }x\}$ is comeagre in $X$. \end{definition} - \begin{definition} Let $X$ be a nonempty topological space and let + \begin{definition} + Let $X$ be a nonempty topological space and let $A\subseteq X$. The \emph{Banach-Mazur game of $A$}, denoted as $G^{\star\star}(A)$ is defined as follows: Players $I$ and $\textit{II}$ take turns in playing nonempty open sets $U_0, V_0, @@ -646,6 +648,56 @@ $\theta[R\upharpoonright_X] = X, \theta[R\upharpoonright_Y] = Y$. \end{proof} + \begin{theorem} + Let $\cC$ be a Fraïssé class of finite structures in a relational language + $L$ of some theory $T$. Let $\cD$ be a class of finite structures of the + theory $T$ in a relational language $L$ 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 the Fraïssé + limit of $\cC$ is isomorphic to the Fraïssé limit of $\cD$ reduced + to the language $L$. + \end{theorem} + + \begin{proof} + Let $\Gamma=\Flim(\cC)$ and $(\Pi, \sigma) =\Flim(\cD)$. By the Fraïssé + theorem \ref{theorem:fraisse_thm} it suffices to show that the age of $\Pi$ + is $\cC$ and that it has the weak ultrahomogeneity in the class $\cC$. The + former comes easily, as for every structure $A\in \cC$ we have the structure + $(A, \id_A)\in \cD$, which means that the structure $A$ embedds into $\Pi$. + Also, if a structure $(B, \beta)\in\cD$ embedds into $\cD$, then $B\in\cC$. + Hence, $\cC$ is indeed the age of $\Pi$. + + Now, take any structure $A, B\in\cC$ such that $A\subseteq B$. Without the + loss of generality assume that $A = B\cap \Pi$. Let $\bar{A}$ be the + smallest structure closed on the automorphism $\sigma$ and containg $A$. + 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: + + \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 in $\cC$. + Hence, it is isomorphic to $\Gamma$. + \end{proof} + \section{Conjugacy classes in automorphism groups} TODO: @@ -659,6 +711,7 @@ $[f]_{G} = \{g\in G\mid f\subseteq g\}$. \subsection{Prototype: pure set} + In this section, $M=(M,=)$ is an infinite countable set (with no structure beyond equality). @@ -881,7 +934,8 @@ Take any finite graph with automorphism $(B, \beta)$. 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 $(i, j) = \min\{\{0,1,\ldots\}\times\bN\setminus X_{n-1}\}$. + By the bookkeeping there was $n$ such that + $(i, j) = \min\{\{0,1,\ldots\}\times\bN\setminus X_{n-1}\}$. This means that $(B, \beta)$ embeds into $(\FrGr_n, g_n)$, hence it embeds into $(\FrGr, g)$, thus it has age $\cH$. With a similar argument we can see that $(\FrGr, g)$ is weakly ultrahomogeneous. @@ -893,5 +947,32 @@ set $A$. \end{proof} + \begin{corollary} + Let $\mathcal{W}$ be a Fraïssé class of finitely generated $L$-structures of + a theory $T$. Let $\mathcal{V}$ be the class of finitely generated structures + of $T$ with an additional unary function interpreted as an automoprphism of + the structure. If $\mathcal{W}$ has weak Hrushovski property and $\mathcal{V}$ + is a Fraïssé class, then $\mathcal{W}$ has a generic automorphism. + \end{corollary} + + TODO: pokazać że w ogólności granica Fraissego V bez tego automorfizmu jest + izomorficzna z W, dopiero wtedy można ten dowód tak uogólnić. + + \begin{proof} + The proof is an abstract version of the theorem for the random graph. + \end{proof} + + \subsection{Properties of the generic automorphism} + + \begin{proposition} + Let $\sigma$ be the generic automorphism of the random graph $\FrGr$. Then + the graph induced by the set of the fixed points of $\sigma$ is isomorphic + to $\FrGr$. + \end{proposition} + + \begin{proof} + Let $F = \{v\in\FrGr\mid \sigma(v) = v\}$. It suffices to show that $F$ is + infinite and has the random graph property. + \end{proof} \printbibliography \end{document} -- cgit v1.2.3