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diff --git a/sections/introduction.tex b/sections/introduction.tex index aedc345..433f3e9 100644 --- a/sections/introduction.tex +++ b/sections/introduction.tex @@ -1,33 +1,42 @@ \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} |