From 43d8381bf8e71d3418f861bc73a8acafa2f25a10 Mon Sep 17 00:00:00 2001
From: Franciszek Malinka <franciszek.malinka@gmail.com>
Date: Tue, 5 Jul 2022 22:16:39 +0200
Subject: Introduction scratch
---
lic_malinka.pdf | Bin 419937 -> 463373 bytes
lic_malinka.tex | 29 +++++++++++++++++++++++++++++
sections/fraisse_classes.tex | 4 ++--
sections/introduction.tex | 30 +++++++++++++++++++++++++++++-
4 files changed, 60 insertions(+), 3 deletions(-)
diff --git a/lic_malinka.pdf b/lic_malinka.pdf
index dc875f1..4fbb6bf 100644
Binary files a/lic_malinka.pdf and b/lic_malinka.pdf differ
diff --git a/lic_malinka.tex b/lic_malinka.tex
index 96f97a3..a469805 100644
--- a/lic_malinka.tex
+++ b/lic_malinka.tex
@@ -113,6 +113,35 @@
\author{Franciszek Malinka}
\begin{document}
+ \newpage
+ \thispagestyle{empty}
+ \begin{center}
+ \textbf{\large Uniwersytet Wrocławski\\
+ Wydział Matematyki i Informatyki\\
+ Instytut Matematyczny}\\
+ \textit{\large specjalność: ISIM}\\
+ \vspace{4cm}
+ \textbf{\textit{\large Franciszek Malinka}}\\
+ \vspace{0.5cm}
+ {\Large Generic automorphisms as Fraïssé limits}\\
+ \end{center}
+ \vspace{3cm}
+ {\large \hspace*{6.5cm}Praca licencjacka\\
+ \hspace*{6.5cm}napisana pod kierunkiem\\
+ \hspace*{6.5cm}dra Tomasza Rzepeckiego}\\
+ \vfill
+ \begin{center}
+ {\large Wrocław 2022}\\
+ \end{center}
+
+ \newpage
+ \thispagestyle{empty}
+ \begin{center}
+ \textit{This page is intentionally left blank}\\
+ \end{center}
+ \vfill
+ \newpage
+
\begin{abstract}
Abstract
\end{abstract}
diff --git a/sections/fraisse_classes.tex b/sections/fraisse_classes.tex
index ca2247e..9110f44 100644
--- a/sections/fraisse_classes.tex
+++ b/sections/fraisse_classes.tex
@@ -9,7 +9,7 @@
\subsection{Definitions}
\begin{definition}
Let $L$ be a signature and $M$ be an $L$-structure. The \emph{age} of $M$ is
- the class $\bK$ of all finitely generated structures that embeds into $M$.
+ the class $\bK$ of all finitely generated structures that embed into $M$.
The age of $M$ is also associated with class of all structures embeddable in
$M$ \emph{up to isomorphism}.
\end{definition}
@@ -63,7 +63,7 @@
\begin{tikzcd}
& D & \\
A \arrow[ur, dashed, "g"] & & B \arrow[ul, dashed, "h"'] \\
- & C \arrow[ur, "f"'] \arrow[ul, "e"]
+ & C \arrow[ur, "f"'] \arrow[ul, "e"] &
\end{tikzcd}
\end{center}
\end{definition}
diff --git a/sections/introduction.tex b/sections/introduction.tex
index 89004c7..aedc345 100644
--- a/sections/introduction.tex
+++ b/sections/introduction.tex
@@ -1,5 +1,33 @@
\documentclass[../lic_malinka.tex]{subfiles}
\begin{document}
- There will be something!
+ Model theory is a field of mathematics that classify and construct
+ structures with particular properties. It desribes classical mathematical
+ objects in a broader context, abstract their properties and study
+ connections between simingly unrelated structures. Roland Fraïssé was
+ French logician who established many important notions in contemporary
+ model theory. He was one of the first to utilize back-and-forth argument,
+ a fundamental model theoretical method in construction of
+ elementary equivalent structures. The Ehrenfeuht-Fraïssé games is a
+ concept that proved useful in classical logic, model theory, but also
+ finite model theory (which is a filed of theoretical informatics rather
+ than mathematics).
+
+ This work study limits of Fraïssé classes with additional combinatorial
+ and categorical properties. The key theorem \ref{theorem:generic_aut_general}
+ says that a Fraïssé class with canonical amalgamation and weak Hrushovsky
+ property has a generic automorphism. This result was known before,
+ for example [DODAC GDZIE TO BYLO...]. However, we show a new way to construct
+ a generic automorphism by extending the structures of the class by an
+ automorphism and considering limit of such extended Fraïssé class. We achieve
+ this by using the Banach-Mazur games, a well known objects of general topology
+ which prove useful in study of comeager sets.
+
+ The prototype structure of the paper is the random graph (also known as the
+ Rado graph), the Fraïssé limit of the class of finite undirected graphs.
+ It serves as a useful example, gives an intuition of the Fraïssé limits,
+ weak Hrushovsky property and free amalgamation.
+
+
+
\end{document}
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