From 22dfe4dade829c5b9fc1f8928f0c78ce99084e19 Mon Sep 17 00:00:00 2001 From: Franciszek Malinka Date: Sun, 29 May 2022 15:20:55 +0200 Subject: Kluczowy dowod uogolniony! --- lic_malinka.pdf | Bin 399632 -> 398521 bytes lic_malinka.tex | 153 +++++++++++++++++++++----------------------------------- 2 files changed, 57 insertions(+), 96 deletions(-) diff --git a/lic_malinka.pdf b/lic_malinka.pdf index a0b2228..64f797a 100644 Binary files a/lic_malinka.pdf and b/lic_malinka.pdf differ diff --git a/lic_malinka.tex b/lic_malinka.tex index 9b218ee..9cce430 100644 --- a/lic_malinka.tex +++ b/lic_malinka.tex @@ -649,6 +649,7 @@ \end{proof} \begin{theorem} + \label{theorem:isomorphic_fr_lims} 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 @@ -779,35 +780,32 @@ $g\circ f_1 = f_2 \circ g$ which exactly means that $f_1, f_2$ conjugate. \end{proof} - - % \begin{proposition} Suppose $\cC$ is a Fraïssé class in a relational - % language with WHP. Then generically, for an $f\in \Aut(\Flim(\cC))$, all - % orbits of $f$ are finite. \end{proposition} \begin{proposition} Suppose - % $\cC$ is a Fraïssé class in an arbitrary countable language with WHP. - % Then generically, for an $f\in \Aut(\Flim(\cC))$ ... \end{proposition} - - % \begin{proposition} Generically, the set of fixed points of $f\in - % \Aut(M)$ is isomorphic to $M$ (as a graph). \end{proposition} - \begin{theorem} - \label{theorem:generic_aut_graph} - Let $\FrGr$ be the Fraïssé limit of the class of all finite graphs $\bK$. - Then $\FrGr$ has a generic automorphism $\tau\in\Aut(\FrGr)$, i.e. the - conjugacy class of $\tau$ is comeagre in $G = \Aut(\FrGr)$. + \label{theorem:generic_aut_general} + Let $\cC$ be a Fraïssé class of finite structures of a theory $T$ in a + relational language $L$. Let $\cD$ be the class of the finite structures of + $T$ in the 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 there is a comeagre conjugacy class in the + automorphism group of the $\Flim(\cC)$. \end{theorem} \begin{proof} + Let $\Gamma = \Flim(\cC)$ and $(\Pi, \sigma) = \Flim(\cD)$. Let $G = \Aut(\Gamma)$, + i.e. $G$ is the automorphism group of $\Gamma$. First, by the theorem + \ref{theorem:isomorphic_fr_lims}, we may assume without the loss of generality + that $\Pi = \Gamma$. We will construct a strategy for the second player in the Banach-Mazur game on the topological space $G$. This strategy will give us a subset - $A\subseteq G$ and as we will see, this will also be a subset of the - conjugacy class of $\tau$. By the Banach-Mazur theorem - \ref{theorem:banach_mazur_thm} this will prove that the class is comeagre. + $A\subseteq G$ and as we will see a subset of a cojugacy class in $G$. + By the Banach-Mazur theorem \ref{theorem:banach_mazur_thm} this will prove + that this class is comeagre. Recall, $G$ has a basis consisting of open sets $\{g\in G\mid g\upharpoonright_A = g_0\upharpoonright_A\}$ for some - finite set $A\subseteq \FrGr$ and some automorphism $g_0\in G$. In other + 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 - isomorphism $g_0$ of $\FrGr$. By $B_{g}\subseteq G$ we denote a basic + isomorphism $g_0$ of $\Gamma$. By $B_{g}\subseteq G$ we denote a basic open subset given by a finite partial isomorphism $g$. From now on we will consider only finite partial isomorphism $g$ such that $B_g$ is nonempty. @@ -819,24 +817,22 @@ the corresponding basic open subsets $B_{f_0}\supseteq B_{g_0}\supseteq\ldots$. Our goal is to choose $g_i$ in such a manner that the resulting function - $g = \bigcap^{\infty}_{i=0}g_i$ will be an automorphism of the random graph - such that $(\FrGr, g) = \Flim{\cH}$, i.e. the Fraïssé limit of finite - graphs with automorphism. Precisely, $\bigcap^{\infty}_{i=0}B_{g_i} = \{g\}$. - By the Fraïssé theorem \ref{theorem:fraisse_thm} - it will follow that $(\FrGr, g)\cong (\FrAut, \sigma)$. By the - proposition \ref{proposition:graph-aut-is-normal} we can assume without - the loss of generality that $\FrAut = \FrGr$ as a plain graph. Hence, + $g = \bigcap^{\infty}_{i=0}g_i$ will be an automorphism of the Fraïssé limit + $\Gamma$ such that $(\Gamma, g) = \Flim{\cD}$. + Precisely, $\bigcap^{\infty}_{i=0}B_{g_i} = \{g\}$, + by the Fraïssé theorem \ref{theorem:fraisse_thm} + it will follow that $(\Gamma, g)\cong (\Pi, \sigma)$. Hence, by the fact \ref{fact:conjugacy}, $g$ and $\sigma$ conjugate in $G$. Once again, by the Fraïssé theorem \ref{theorem:fraisse_thm} and the \ref{lemma:weak_ultrahom} lemma constructing $g_i$'s in a way such that - age of $(\FrGr, g)$ is exactly $\cH$ and so that it is weakly ultrahomogeneous - will produce expected result. First, let us enumerate all pairs of finite - graphs with automorphism $\{\langle(A_n, \alpha_n), (B_n, \beta_n)\rangle\}_{n\in\bN}$ + age of $(\Gamma, g)$ is exactly $\cD$ and so that it is weakly ultrahomogeneous + will produce expected result. First, let us enumerate all pairs of structures + $\{\langle(A_n, \alpha_n), (B_n, \beta_n)\rangle\}_{n\in\bN},\in\cD$ such that the first element of the pair embeds by inclusion in the second, i.e. $(A_n, \alpha_n)\subseteq (B_n, \beta_n)$. Also, it may be that - $A_n$ is an empty graph. We enumerate the vertices of the random graph - $\FrGr = \{v_0, v_1, \ldots\}$. + $A_n$ is an empty. We enumerate the elements of the Fraïssé limit + $\Gamma = \{v_0, v_1, \ldots\}$. Fix a bijection $\gamma\colon\bN\times\bN\to\bN$ such that for any $n, m\in\bN$ we have $\gamma(n, m) \ge n$. This bijection naturally induces @@ -850,15 +846,15 @@ such that following properties hold: \begin{enumerate}[label=(\roman*)] - \item $g_n$ is an automorphism of the induced subgraph $\FrGr_n$, + \item $g_n$ is an automorphism of the induced substructure $\Gamma_n$, \item $g_n(v_n)$ and $g_n^{-1}(v_n)$ are defined, \item let $\{\langle (A_{n,k}, \alpha_{n, k}), (B_{n,k}, \beta_{n,k}), f_{n, k}\rangle\}_{k\in\bN}$ - be the enumeration of all pairs of finite graphs with automorphism such - that the first is a substructure of the second, i.e. + be the enumeration of all pairs of finite structures of $T$ with automorphism + such that the first is a substructure of the second, i.e. $(A_{n,k}, \alpha_{n,k})\subseteq (B_{n,k}, \beta_{n,k})$, and $f_{n,k}$ is an embedding of $(A_{n,k}, \alpha_{n,k})$ in the $\FrGr_{n-1}$ (which - is the graph induced by $g_{n-1}$). Let + is the substructure induced by $g_{n-1}$). Let $(i, j) = \min\{(\{0, 1, \ldots\} \times \bN) \setminus X_{n-1}\}$ (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 @@ -866,79 +862,59 @@ \begin{center} \begin{tikzcd} - & (\FrGr_n, g_n) & \\ + & (\Gamma_n, g_n) & \\ (B_{i,j}, \beta_{i,j}) \arrow[ur, dashed, "\hat{f}_{i,j}"] & & (\FrGr_{n-1}, g_{n-1}) \arrow[ul, dashed, "\subseteq"'] \\ & (A_{i,j}, \alpha_{i,j}) \arrow[ur, "f_{i,j}"'] \arrow[ul, "\subseteq"] \end{tikzcd} \end{center} - - % \begin{center} - % \begin{tikzcd} - % (A_{i,j}, \alpha_{i,j}) \arrow[d, "\subseteq"'] \arrow[r, "f_{i,j}"] & (\FrGr_{n-1}, g_{n-1}) \arrow[d, "\subseteq"] \\ - % (B_{i,j}, \beta_{i,j}) \arrow[r, dashed, "\hat{f}_{i,j}"'] & (\FrGr_n, g_n) - % \end{tikzcd} - % \end{center} \end{enumerate} - First item makes sure that no infinite orbit will not be present in $g$. The + First item makes sure that no infinite orbit will be present in $g$. The second item together with the first one are necessary for $g$ to be an - automorphism of $\FrGr$. The third item is the one that gives weak + automorphism of $\Gamma$. The third item is the one that gives weak ultrahomogeneity. Now we will show that indeed such $g_n$ may be constructed. - First, we will suffice the item (iii). Namely, we will construct $\FrGr'_n, g'_n$ + First, we will suffice the item (iii). Namely, we will construct $\Gamma'_n, g'_n$ such that $g_{n-1}\subseteq g'_n$ and $f_{i,j}$ extends to an embedding of - $(B_{i,j}, \beta_{i,j})$ to $(\FrGr'_n, g'_n)$. But this can be easily - done by the fact, that $\cH$ has the amalgamation property. Moreover, without + $(B_{i,j}, \beta_{i,j})$ to $(\Gamma'_n, g'_n)$. But this can be easily + done by the fact, that $\cD$ has the amalgamation property. Moreover, without the loss of generality we can assume that all embeddings are inclusions. \begin{center} \begin{tikzcd} - & (\FrGr'_n, g'_n) & \\ - (B_{i,j}, \beta_{i,j}) \arrow[ur, dashed, "\subseteq"] & & (\FrGr_{n-1}, g_{n-1}) \arrow[ul, dashed, "\subseteq"'] \\ + & (\Gamma'_n, g'_n) & \\ + (B_{i,j}, \beta_{i,j}) \arrow[ur, dashed, "\subseteq"] & & (\Gamma_{n-1}, g_{n-1}) \arrow[ul, dashed, "\subseteq"'] \\ & (A_{i,j}, \alpha_{i,j}) \arrow[ur, "\subseteq"'] \arrow[ul, "\subseteq"] \end{tikzcd} \end{center} - % Without the loss of generality - % we may assume that $f_{i,j}$ is an inclusion and that $A_{i,j} = B_{i,j}\cap\FrGr_{n-1}$. - % Then let $\FrGr'_n = B_{i,j}\cup\FrGr_{n-1}$ and $g'_n = g_{n-1}\cup \beta_{i,j}$. - % Then $(B_{i,j}, \beta_{i,j})$ simply embeds by inclusion in $(\FrGr'_n, g'_n)$, - % i.e. this diagram commutes: - - % \begin{center} - % \begin{tikzcd} - % (A_{i,j}, \alpha_{i,j}) \arrow[d, "\subseteq"'] \arrow[r, "\subseteq"] & (\FrGr_{n-1}, g_{n-1}) \arrow[d, "\subseteq"] \\ - % (B_{i,j}, \beta_{i,j}) \arrow[r, dashed, "\subseteq"'] & (\FrGr'_n, g'_n) - % \end{tikzcd} - % \end{center} - - % The argument that those are actually embeddings is almost the same as in - % proof of the amalgamation property of $\cH$. - - It may be also assumed without the loss of generality that $\FrGr'_n\subseteq \FrGr$. - Of course by the recursive assumption $\FrGr_{n-1}\subseteq\FrGr$. The - $\FrGr'_n \setminus\FrGr_{n-1} = B_{i,j}\setminus A_{i,j}$ can be found in - $\FrGr$ by the random graph property -- we can find vertices of the remaining - part of $B_{i,j}$ each at a time so that all edges are correct. - - Now, by the WHP of $\bK$ we can extend the graph $\FrGr'_n\cup\{v_n\}$ together - with its partial isomorphism $g'_n$ to a graph $\FrGr_n$ together with its + By the weak ultrahomogeneity we may assume that $\Gamma'_n\subseteq \Gamma$: + + \begin{center} + \begin{tikzcd} + (B_{i,j}\cup\Gamma_{n-1}, \beta_{i,j}\cup g_{n-1}) \arrow[d, "\subseteq"'] \arrow[r, "\subseteq"] & \Gamma \\ + (\Gamma'_{n}, g'_n)\arrow[ur, dashed, "f"'] + \end{tikzcd} + \end{center} + + Now, by the WHP of $\bK$ we can extend the graph $\Gamma'_n\cup\{v_n\}$ together + with its partial isomorphism $g'_n$ to a graph $\Gamma_n$ together with its automorphism $g_n\supseteq g'_n$ and without the loss of generality we - may assume that $\FrGr_n\subseteq\FrGr$. This way we've constructed $g_n$ + may assume that $\Gamma_n\subseteq\Gamma$. This way we've constructed $g_n$ that has all desired properties. Now we need to see that $g = \bigcap^{\infty}_{n=0}g_n$ is indeed an automorphism - of $\FrGr$ such that $(\FrGr, g)$ has the age $\cH$ and is weakly ultrahomogeneous. - It is of course an automorphism of $\FrGr$ as it is defined for every $v\in\FrGr$ + of $\Gamma$ such that $(\Gamma, g)$ has the age $\cH$ and is weakly ultrahomogeneous. + It is of course an automorphism of $\Gamma$ as it is defined for every $v\in\Gamma$ and is a sum of increasing chain of finite automorphisms. - Take any finite graph with automorphism $(B, \beta)$. Then, there are + Take any finite structure of $T$ 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 $(i, j) = \min\{\{0,1,\ldots\}\times\bN\setminus X_{n-1}\}$. - 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. + This means that $(B, \beta)$ embeds into $(\Gamma_n, g_n)$, hence it embeds + into $(\Gamma, g)$, thus it has age $\cH$. + With a similar argument we can see that $(\Gamma, g)$ is weakly ultrahomogeneous. By this we know that $g$ and $\sigma$ conjugate in $G$. As we stated in the beginning of the proof, the set $A$ of possible outcomes of the game (i.e. @@ -947,21 +923,6 @@ 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} -- cgit v1.2.3