diff options
Diffstat (limited to 'sections')
| -rw-r--r-- | sections/fraisse_classes.tex | 4 | ||||
| -rw-r--r-- | sections/introduction.tex | 30 |
2 files changed, 31 insertions, 3 deletions
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} |