From 29fb1dc0cb80c83f071079009ba487720685f05a Mon Sep 17 00:00:00 2001 From: Franciszek Malinka Date: Tue, 3 May 2022 13:12:54 +0200 Subject: poprawki do kluczowego dowodu --- lic_malinka.pdf | Bin 391744 -> 391756 bytes lic_malinka.tex | 81 ++++++++++++++++++++++++++++++++------------------------ 2 files changed, 47 insertions(+), 34 deletions(-) diff --git a/lic_malinka.pdf b/lic_malinka.pdf index 2c4447f..edf6b66 100644 Binary files a/lic_malinka.pdf and b/lic_malinka.pdf differ diff --git a/lic_malinka.tex b/lic_malinka.tex index fe992a1..5e52034 100644 --- a/lic_malinka.tex +++ b/lic_malinka.tex @@ -663,6 +663,7 @@ beyond equality). \begin{proposition} + \label{proposition:cojugate-classes} If $f_1,f_2\in \Aut(M)$, then $f_1$ and $f_2$ are conjugate if and only if for each $n\in \bN\cup \{\aleph_0\}$, $f_1$ and $f_2$ have the same number of orbits of size $n$. @@ -703,20 +704,11 @@ is an open dense set. It is a sum over basic open sets generated by finite permutations with $m$ in their domain. Denseness is also easy to see. - Finally, we can say that + Finally, by the proposition \ref{proposition:cojugate-classes}, we can say that $$\sigma^{\Aut(M)}=\bigcap_{n=1}^\infty A_n \cap \bigcap_{m\in M} B_m,$$ which concludes the proof. - \end{proof} - % \begin{proposition} - % An automorphism $f$ of $M$ is generic if and only if... - % \end{proposition} - - % \begin{proof} - - % \end{proof} - \subsection{More general structures} \begin{fact} @@ -821,10 +813,18 @@ \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) + & (\FrGr_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 @@ -832,23 +832,35 @@ automorphism of $\FrGr$. 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 (iii) item. Namely, we will construct $\FrGr'_n, g'_n$ + First, we will suffice the item (iii). Namely, we will construct $\FrGr'_n, g'_n$ such that $g_{n-1}\subseteq g'_n$ and $f_{i,j}$ extends to an embedding of - $(B_{i,j}, \beta_{i,j})$ to $(\FrGr'_n, g'_n)$. 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 $\cH$. + $(B_{i,j}, \beta_{i,j})$ to $(\FrGr'_n, g'_n)$. But this can be easily + done by the fact, that $\cH$ has the amalgamation property. Moreover, without + 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"'] \\ + & (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 @@ -858,12 +870,12 @@ 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 - automorphism $g_n\supseteq g'_n$, where without the loss of generality we + 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 $g_n$ 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 has weak ultrahomogeneity. + 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 $v\in\FrGr$ and is a sum of increasing chain of finite automorphisms. @@ -871,13 +883,14 @@ $i, j$ 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 $(\FrGr, g)$. With a similar argument we can see that $(\FrGr, g)$ is - weakly ultrahomogeneous. + into $(\FrGr, g)$, thus it has age $\cH$. + With a similar argument we can see that $(\FrGr, g)$ 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 of possible outcomes of the game (i.e. + beginning of the proof, the set $A$ of possible outcomes of the game (i.e. possible $g$'s we end up with) is comeagre in $G$, thus $\sigma^G$ is also - comeagre and $\sigma$ is a generic automorphism. + comeagre and $\sigma$ is a generic automorphism, as it contains a comeagre + set $A$. \end{proof} \printbibliography -- cgit v1.2.3