From 15f7c05afe4f41c4debf9400a5d84c6fb502ad4b Mon Sep 17 00:00:00 2001
From: Franciszek Malinka <franciszek.malinka@gmail.com>
Date: Sun, 21 Aug 2022 21:57:18 +0200
Subject: Examples introduced

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+\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}
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