Zaczątki klasy fraissego dla grafow z automorfizmem

2 files changed, 33 insertions, 1 deletions

diff --git a/lic_malinka.pdf b/lic_malinka.pdf
index 22a9c6a..6f1f6c1 100644
--- a/lic_malinka.pdf
+++ b/lic_malinka.pdf
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}