From 3e1cedd7a778548ab15148ce820e7f90620a5af4 Mon Sep 17 00:00:00 2001
From: Franciszek Malinka <franciszek.malinka@gmail.com>
Date: Sun, 27 Mar 2022 20:45:01 +0200
Subject: =?UTF-8?q?Zacz=C4=85tki=20klasy=20fraissego=20dla=20grafow=20z=20?=
 =?UTF-8?q?automorfizmem?=
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---
 lic_malinka.tex | 34 +++++++++++++++++++++++++++++++++-
 1 file changed, 33 insertions(+), 1 deletion(-)

(limited to 'lic_malinka.tex')

diff --git a/lic_malinka.tex b/lic_malinka.tex
index ede7008..6200d4c 100644
--- a/lic_malinka.tex
+++ b/lic_malinka.tex
@@ -51,13 +51,15 @@
 \newcommand{\cV}{\mathcal{V}} 
 \newcommand{\cU}{\mathcal{U}}
 \newcommand{\cG}{\mathcal{G}}
+\newcommand{\cH}{\mathcal{H}}
 \newcommand{\bN}{\mathbb N} 
 \newcommand{\bR}{\mathbb R}
 \newcommand{\bZ}{\mathbb Z} 
 \newcommand{\bQ}{\mathbb Q}
-\newcommand{\bK}{\mathbb K}
+\newcommand{\bK}{\mathcal K}
 
 \DeclareMathOperator{\im}{{Im}} 
+\DeclareMathOperator{\id}{{id}} 
 \DeclareMathOperator{\lin}{{Lin}}
 \DeclareMathOperator{\Th}{{Th}}
 
@@ -544,6 +546,36 @@
     It may be there some day, but it may not!
   \end{proof}
 
+  \subsection{Graphs with automorphism}
+  The language and theory of unordered graphs is fairly simple. We extend the
+  language by one unary symbol $\sigma$ and interpret it as an arbitrary
+  automorphism on the graph structure. It turns out that the class of such
+  structures forms a Fraïssé class.
+
+  \begin{proposition}
+    Let $\cH$ be the class of all finite graphs with automorphism, i.e.
+    structures in the language $(E, \sigma)$ such that $E$ is a symmetric
+    relation and $\sigma$ is an automorphism on the structure. $\cH$ is
+    a Fraïssé class.
+  \end{proposition}
+  \begin{proof}
+    Countability and HP are obivous, JEP follows by the same argument as in
+    plain graphs. We need to show that the class has the amalgamation property.
+
+    Take any graphs $(A, \alpha), (B, \beta), (C,\gamma)$ such that $A$ embedds
+    into $B$ and $C$. Let $D$ be the amalgamation of $B$ and $C$ over $A$ as in
+    the proof for the plain graphs. We will define the automorphis
+    $\delta\in\Aut(D)$ so it extends $\beta$ and $\gamma$. (TODO: chyba nie
+    tylko extends ale coś więcej: wiem o co chodzi, ale nie wiem jak to
+    napisać) We let $\delta_{\upharpoonright X} = \id_X$ for $X\in \{A,
+    B\setminus A, C\setminus B\}$. Let's check the definition is correct. In
+    order to do that, we have to show that for any $u, v\in
+    D (uE_Dv\leftrightarrow \delta(u)E_D\delta(v))$. We have a few cases:
+    \begin{itemize}
+      \item 
+    \end{itemize}
+  \end{proof}
+
   \section{Conjugacy classes in automorphism groups}
 
   \subsection{Prototype: pure set} 
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