From a15a0040023eb4f8f2b9d9653789063b86ccbe62 Mon Sep 17 00:00:00 2001 From: Franciszek Malinka Date: Wed, 24 Aug 2022 23:07:30 +0200 Subject: Extended introducion --- sections/examples.tex | 9 ++++----- sections/introduction-pl.tex | 7 +++++-- sections/introduction.tex | 9 ++++++--- 3 files changed, 15 insertions(+), 10 deletions(-) (limited to 'sections') diff --git a/sections/examples.tex b/sections/examples.tex index 98a193b..fdb8c6a 100644 --- a/sections/examples.tex +++ b/sections/examples.tex @@ -160,11 +160,10 @@ is a Fraïssé class with WHP and CAP. \end{example} - The prove of this is relatively easy, knowing that there is essentially one - vector space of every finite dimension and that every linear independent subset - of a vector space can be extended to a basis of this space. The Fraïssé limit - of $\cV$ is the $\omega$-dimensional vector space. Thus, by our key Theorem - \ref{theorem:key-theorem} we know that it has a generic automorphism. + Vector spaces of the same dimension are isomorphic, thus it is obvious that + $\cV$ is essentially countable. Also $HP$ and $JEP$ are obvious, as we can + always embed space with smaller dimension into the bigger one. Amalgamation + works exactly the same. In fact, such amalgamation is indeed canonical. \begin{example} The class of all finite graphs $\cG$ is a Fraïssé class with WHP and free diff --git a/sections/introduction-pl.tex b/sections/introduction-pl.tex index 7e8243e..dca6599 100644 --- a/sections/introduction-pl.tex +++ b/sections/introduction-pl.tex @@ -38,6 +38,9 @@ grami Banacha-Mazura, które są dobrze znanym narzędziem w deskryptywnej teorii mnogości. - % Opisana konstrukcja generycznego automorfizmu okazuje się pomocna w dowodzeniu - % niektórych własności tego automorfizmu (patrz \ref{proposition:fixed_points}). + Opisana konstrukcja generycznego automorfizmu okazuje się pomocna w dowodzeniu + niektórych własności tego automorfizmu (patrz \ref{proposition:fixed_points}). + W ostatnim rozdziale przytaczamy przykłady klas Fraïsségo ze słabą własnością + Hrushovskiego i kanoniczną amalgamacją oraz charakteryzujemy ich granice + oraz generyczny automorfizm. \end{document} diff --git a/sections/introduction.tex b/sections/introduction.tex index 8886847..57b8de1 100644 --- a/sections/introduction.tex +++ b/sections/introduction.tex @@ -37,7 +37,10 @@ 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)). + 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}). In the last section we give examples + and anti-examples of Fraïssé classes with weak Hrushovski property and + canonical amalgamation, characterize Fraïssé limits and generic automorphism + of these classes. \end{document} -- cgit v1.2.3