From bd01da032991f9671557ef64e23ca684fa6c995a Mon Sep 17 00:00:00 2001
From: Franciszek Malinka <franciszek.malinka@gmail.com>
Date: Sat, 9 Jul 2022 13:44:47 +0200
Subject: Introduction corrected
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\documentclass[../lic_malinka.tex]{subfiles}
\begin{document}
- 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).
+ Model theory is a field of mathematics that classifies and constructs
+ structures with particular properties (particularly those expressible
+ in first order logic). It describes classical mathematical
+ objects in a broader context, abstracts their properties and studies
+ connections between seemingly unrelated structures.
+ This work studies limits of Fraïssé classes with additional combinatorial
+ and categorical properties. Fraïssé classes are frequently used in model
+ theory,
+ both as a source of examples and to analyse particular ``generic'' structures.
- 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 notion of Fraïssé class and its limit is
+ due to the French logician Roland Fraïssé. He also introduced the
+ back-and-forth argument, a fundamental model theoretical method in
+ construction of elementarily equivalent structures, upon which
+ Ehrenfeucht-Fraïssé games are based.
- The prototype structure of the paper is the random graph (also known as the
+ The prototypical example for this paper is the random graph
+ \ref{definition: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.
+ weak Hrushovski property and free amalgamation. Perhaps most importantly,
+ the random graph has a so-called generic automorphism (DODAĆ DEFINICJĘ
+ GENERYCZNEGO AUTOMORFIZMU I ZLINKOWAĆ TUTAJ), which was first proved
+ by Truss in \cite{truss_gen_aut}, where he also introduced the term.
-
+ The key theorem \ref{theorem:generic_aut_general}
+ says that a Fraïssé class with canonical amalgamation and weak Hrushovski
+ property has a generic automorphism. The fact that such an automorphism
+ exists in this case follows from the classical results of Ivanov \cite{ivanov_1999}
+ and Kechris-Rosendal \cite{https://doi.org/10.1112/plms/pdl007}, we show a new way to construct
+ a generic automorphism by expanding the structures of the class by a (total)
+ automorphism and considering limit of such extended Fraïssé class. We achieve
+ this by using the Banach-Mazur games, a well known method in the descriptive
+ set theory, which proves useful in the study of comeagre sets.
+ Finally, we show how this construction of the generic automorphism can be
+ used to deduce some properties of generic automorphisms
+ (see \ref{proposition:fixed_points}, (COŚ JESZCE)).
\end{document}
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