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author | Franciszek Malinka <franciszek.malinka@gmail.com> | 2022-07-05 22:16:39 +0200 |
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committer | Franciszek Malinka <franciszek.malinka@gmail.com> | 2022-07-05 22:16:39 +0200 |
commit | 43d8381bf8e71d3418f861bc73a8acafa2f25a10 (patch) | |
tree | 29a58389574db77133507d374e80470229d7eed5 /sections/introduction.tex | |
parent | 14de439d475ec315ae40c85f813036e5568a019f (diff) |
Introduction scratch
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-rw-r--r-- | sections/introduction.tex | 30 |
1 files changed, 29 insertions, 1 deletions
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} |