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

---
 sections/fraisse_classes.tex |  4 ++--
 sections/introduction.tex    | 30 +++++++++++++++++++++++++++++-
 2 files changed, 31 insertions(+), 3 deletions(-)

(limited to 'sections')

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}
-- 
cgit v1.2.3