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authorFranciszek Malinka <franciszek.malinka@gmail.com>2022-08-21 21:57:18 +0200
committerFranciszek Malinka <franciszek.malinka@gmail.com>2022-08-21 21:57:18 +0200
commit15f7c05afe4f41c4debf9400a5d84c6fb502ad4b (patch)
tree46a1ad5bcfeae2bae5857dfd1e4ee9993b542c57
parentef3fc25228cc13c9f39192813653b1b6dc339406 (diff)
Examples introduced
-rw-r--r--lic_malinka.pdfbin479056 -> 487343 bytes
-rw-r--r--lic_malinka.tex5
-rw-r--r--sections/examples.tex120
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diff --git a/lic_malinka.pdf b/lic_malinka.pdf
index 3725a5b..961a93b 100644
--- a/lic_malinka.pdf
+++ b/lic_malinka.pdf
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diff --git a/lic_malinka.tex b/lic_malinka.tex
index 6cfaac4..47e1499 100644
--- a/lic_malinka.tex
+++ b/lic_malinka.tex
@@ -23,6 +23,7 @@
\usepackage{tikz-cd}
\usepackage{tikz}
\usetikzlibrary{arrows,arrows.meta}
+\usetikzlibrary{shapes.geometric}
\tikzcdset{arrow style=tikz, diagrams={>=latex}}
\usepackage{etoolbox}
@@ -60,6 +61,7 @@
\newcommand{\bZ}{\mathbb Z}
\newcommand{\bQ}{\mathbb Q}
\newcommand{\cK}{\mathcal K}
+\newcommand{\cL}{\mathcal L}
\newcommand{\FrAut}{\Pi}
\newcommand{\FrGr}{\Gamma}
@@ -160,5 +162,8 @@
\section{Conjugacy classes in automorphism groups}
\subfile{sections/conj_classes}
+ \section{Examples}
+ \subfile{sections/examples.tex}
+
\printbibliography
\end{document}
diff --git a/sections/examples.tex b/sections/examples.tex
new file mode 100644
index 0000000..0b45b32
--- /dev/null
+++ b/sections/examples.tex
@@ -0,0 +1,120 @@
+\documentclass[../lic_malinka.tex]{subfiles}
+
+\begin{document}
+ \begin{example}
+ Let $\cL$ be the class of all finite linear orderings. Then:
+ \begin{enumerate}
+ \item $\cL$ is a Fraïssé class.
+ \item $\cL$ has canonical amalgamation.
+ \item $\cL$ does not have WHP.
+ \end{enumerate}
+ \end{example}
+
+ $\cL$ of course has HP and is essentialy countable. JEP is also easy,
+ as having two finite linear orderings we can just embed the one with less
+ elements into the bigger one.
+
+ We will show that $\cL$ has CAP. Let $C$ be a finite linear ordered set.
+ We will define $\otimes_C$. Let $A$, $B$ be finite linear orderings that
+ $C$ embeds into. We may suppose that $C = A\cap B$. Then we define an
+ ordering on $D = A\cup B$. For $d,e \in D$, let $d\le_D e$ if one of the
+ following hold:
+ \begin{itemize}
+ \item $d,e\in A$ and $d\le_A e$,
+ \item $d,e\in B$ and $d\le_B e$,
+ \item $d\in A, e\in B$ and there is $c\in C$ such that $d\le_A c$ and
+ $c\le_B e$,
+ \item $d\in B, e\in A$ and there is $c\in C$ such that $d\le_B c$ and
+ $c\le_A e$,
+ \item $d\in A, e\in B$ and for all $c\in C$ $d\le_A c\Leftrightarrow e\le_B c$
+ \end{itemize}
+
+ One can imagine that $D$ can be constructed by laying elements of $C$
+ in a row and putting elements of $A$ and $B$ appropriately between elements
+ of $C$ with all elements of $A$ to the left and all elements of $B$ to
+ the right between two adjacent elements of $C$. This clearly is
+ a canonical amalgamation.
+
+ \vspace{0.5cm}
+ \begin{figure}[h]
+ \centering
+ \begin{tikzpicture}
+ \draw[->,thick] (-1.5,0)--(1.5,0) node[right]{$C$};
+ \foreach \x in {0,...,2}
+ \node at ({-1.5+3/4*(1+\x)},0)[circle,fill=green,inner sep=2pt]{};
+
+ \draw[->] (-1,0.5) -- (-3,1.5);
+ \draw[->] (1,0.5) -- (3,1.5);
+
+ \draw[->,thick] (-6, 2)--(0,2) node[right]{$A$};
+ \foreach \x in {0,...,7}
+ \node at ({-6 + 6/9 * (\x+1)},2)[rectangle,fill=magenta, inner sep=2pt]{};
+ \foreach \x in {2,5,6}
+ \node at ({-6 + 6/9 * (\x+1)},2)[circle,fill=green, inner sep=2pt]{};
+
+ \draw[->] (-3,2.5) -- (-2,3.5);
+
+ \draw[->,thick] (1, 2)--(6,2) node[right]{$B$};
+ \foreach \x in {0,2,4,6}
+ \node at ({1+5/8*(\x+1)},2)[star,fill=blue, inner sep=2pt]{};
+ \foreach \x in {1, 3, 5}
+ \node at ({1+5/8*(\x+1)},2)[circle,fill=green, inner sep=2pt]{};
+
+ \draw[->] (3,2.5) -- (2,3.5);
+
+ \draw[->,thick] (-5, 4)--(5,4) node[right]{$D$};
+ % \foreach \x in {0,...,11}
+ % \node at ({-5+10/13*(\x+1)}, 4)[circle,fill=black,inner sep=2pt]{};
+ \foreach \x in {3,7,9}
+ \node at ({-5+10/13*(\x+1)}, 4)[circle,fill=green,inner sep=2pt]{};
+ \foreach \x in {0,1,4,5,10}
+ \node at ({-5+10/13*(\x+1)}, 4)[rectangle,fill=magenta,inner sep=2pt]{};
+ \foreach \x in {2,6,8,11}
+ \node at ({-5+10/13*(\x+1)}, 4)[star,fill=blue,inner sep=2pt]{};
+ \end{tikzpicture}
+ \caption{Visual representation of the construction.}
+ \end{figure}
+ \vspace{0.5cm}
+
+ On the other hand $\cL$ cannot have $WHP$. This follows from the fact that
+ that only automoprhism of a finite linear ordering is identity, so
+ we cannot extend a partial automoprhism sending exactly one element to
+ some distinct element.
+
+ \begin{definition}
+ Let $X$ be a set. A ternarny relation $\le^C \subseteq X^3$ is a
+ \emph{cyclic order}, where we denote $(a,b,c)\in\le^C$ as $a\le^{C}_{b}c$
+ (or simply $a\le_bc$ when there's only one relation in the context),
+ when it satisfies the following properties:
+ \begin{itemize}
+ \item If $a\le_bc$, then $b\le_ca$.
+ \item If $a\le_bc$, then not $c\le_ba$.
+ \item If $a\le_bc$ and $a\le_cd$, then $a\le_bd$.
+ \item If $a,b,c$ are pairwise distinct, then either $a\le_bc$ or
+ $c\le_ba$.
+ \end{itemize}
+ \end{definition}
+
+ \begin{example}
+ Let $\cC$ be the class of all finite cyclic orderings. Then:
+ \begin{enumerate}
+ \item $\cC$ is a Fraïssé class.
+ \item $\cC$ does not have WHP.
+ \item $\cC$ does not have CAP.
+ \end{enumerate}
+ \end{example}
+
+ \begin{example}
+ The class of all finitely generated vector spaces $\cV$ is a Fraïssé class
+ with WHP and CAP.
+ \end{example}
+ \begin{example}
+ The class of all finite graphs $\cG$ is a
+ \end{example}
+ \begin{example}
+ Graphs without triangles.
+ \end{example}
+ \begin{example}
+ Graphs without 3-paths.
+ \end{example}
+\end{document}