From 930d96d3091323e34a9c35dbc58d377666c3311e Mon Sep 17 00:00:00 2001 From: Franciszek Malinka Date: Sat, 26 Mar 2022 15:25:46 +0100 Subject: =?UTF-8?q?Zmiana=20kolejno=C5=9Bci=20ma=C5=82a?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- lic_malinka.tex | 125 ++++++++++++++++++++++++++++---------------------------- 1 file changed, 63 insertions(+), 62 deletions(-) (limited to 'lic_malinka.tex') diff --git a/lic_malinka.tex b/lic_malinka.tex index d98ecee..ede7008 100644 --- a/lic_malinka.tex +++ b/lic_malinka.tex @@ -455,67 +455,6 @@ % $\Flim(\bK)$ is $\aleph_0$-categorical and has quantifier elimination. % \end{fact} - \section{Conjugacy classes in automorphism groups} - - \subsection{Prototype: pure set} - In this section, $M=(M,=)$ is an infinite countable set (with no structure - beyond equality). - \begin{proposition} - 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 $n$. - \end{proposition} - - \begin{proposition} The conjugacy class of $f\in \Aut(M)$ is dense if - and only if... \end{proposition} \begin{proposition} If $f\in - \Aut(M)$ has an infinite orbit, then the conjugacy class of $f$ is - meagre. - \end{proposition} - - % \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} - Suppose $M$ is an arbitrary structure and $f_1,f_2\in \Aut(M)$. - Then $f_1$ and $f_2$ are conjugate if and only if $(M,f_1)\cong - (M,f_2)$ as structures with one additional unary relation that is - an automorphism. - \end{fact} - - \begin{proof} - Suppose that $f_1 = g^{-1}f_2g$ for some $g\in \Aut(M)$. - Then $g$ is the automorphism we're looking for. On the other hand if - $g\colon (M, f_1)\to (M, f_2)$ is an isomorphism, then - $g\circ f_1 = f_2 \circ g$ which exactly means that $f_1, f_2$ conjugate. - \end{proof} - - \begin{definition} We say that a Fraïssé class $\bK$ has \emph{weak - Hrushovski property} (\emph{WHP}) if for every $A\in \bK$ and an isomorphism - of substructures of $A$ $p\colon A\to A$, there is some $B\in \bK$ such - that $p$ can be extended to an automorphism of $B$, i.e.\ there is an - embedding $i\colon A\to B$ and a $\bar p\in \Aut(B)$ such that the following - diagram commutes: - \begin{center} - \begin{tikzcd} - B\ar[r,dashed,"\bar p"]&B\\ - A\ar[u,dashed,"i"]\ar[r,"p"]&A\ar[u,dashed,"i"] - \end{tikzcd} - \end{center} - \end{definition} - - % \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} - \subsection{Random graph} In this section we'll take a closer look on a class of finite graphs, which @@ -605,7 +544,69 @@ It may be there some day, but it may not! \end{proof} - % \begin{proposition} Generically, the set of fixed points of $f\in + \section{Conjugacy classes in automorphism groups} + + \subsection{Prototype: pure set} + In this section, $M=(M,=)$ is an infinite countable set (with no structure + beyond equality). + \begin{proposition} + 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 $n$. + \end{proposition} + + \begin{proposition} The conjugacy class of $f\in \Aut(M)$ is dense if + and only if... \end{proposition} \begin{proposition} If $f\in + \Aut(M)$ has an infinite orbit, then the conjugacy class of $f$ is + meagre. + \end{proposition} + + % \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} + Suppose $M$ is an arbitrary structure and $f_1,f_2\in \Aut(M)$. + Then $f_1$ and $f_2$ are conjugate if and only if $(M,f_1)\cong + (M,f_2)$ as structures with one additional unary relation that is + an automorphism. + \end{fact} + + \begin{proof} + Suppose that $f_1 = g^{-1}f_2g$ for some $g\in \Aut(M)$. + Then $g$ is the automorphism we're looking for. On the other hand if + $g\colon (M, f_1)\to (M, f_2)$ is an isomorphism, then + $g\circ f_1 = f_2 \circ g$ which exactly means that $f_1, f_2$ conjugate. + \end{proof} + + \begin{definition} We say that a Fraïssé class $\bK$ has \emph{weak + Hrushovski property} (\emph{WHP}) if for every $A\in \bK$ and an isomorphism + of substructures of $A$ $p\colon A\to A$, there is some $B\in \bK$ such + that $p$ can be extended to an automorphism of $B$, i.e.\ there is an + embedding $i\colon A\to B$ and a $\bar p\in \Aut(B)$ such that the following + diagram commutes: + \begin{center} + \begin{tikzcd} + B\ar[r,dashed,"\bar p"]&B\\ + A\ar[u,dashed,"i"]\ar[r,"p"]&A\ar[u,dashed,"i"] + \end{tikzcd} + \end{center} + \end{definition} + + % \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} + \printbibliography \end{document} -- cgit v1.2.3