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Diffstat (limited to 'sections')
| -rw-r--r-- | sections/conj_classes.tex | 2 | ||||
| -rw-r--r-- | sections/examples.tex | 73 |
2 files changed, 62 insertions, 13 deletions
diff --git a/sections/conj_classes.tex b/sections/conj_classes.tex index d37afd5..87fa416 100644 --- a/sections/conj_classes.tex +++ b/sections/conj_classes.tex @@ -144,7 +144,7 @@ such that the following properties hold: \begin{enumerate}[label=(\roman*)] - \item $g_n$ is is a partial automorphism of $\Gamma$ and an automorphism of + \item $g_n$ is a partial automorphism of $\Gamma$ and an automorphism of finitely generated substructure $\Gamma_n$, \item $g_n(v_n)$ and $g_n^{-1}(v_n)$ are defined. \end{enumerate} diff --git a/sections/examples.tex b/sections/examples.tex index 0b45b32..f65ebce 100644 --- a/sections/examples.tex +++ b/sections/examples.tex @@ -79,7 +79,8 @@ 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. + some distinct element. However, in this case, generic automoprhism exists + which was shown by Truss \cite{truss_gen_aut}. \begin{definition} Let $X$ be a set. A ternarny relation $\le^C \subseteq X^3$ is a @@ -95,22 +96,70 @@ \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} + It is easy to visualise a cyclic ordering as a directed (\textit{nomen omen}) + cycle. For example, a 11-element cyclic order could be drawn like this: + + \begin{figure}[h] + \centering + \begin{tikzpicture} + \def \n {11} + \def \radius {3cm} + \def \margin {10} + + \foreach \s in {1,...,\n} + { + \node[draw, circle] at ({360/\n * (\s - 1)}:\radius) {}; + \draw[->, >=latex] ({360/\n * (\s - 1)+\margin}:\radius) + arc ({360/\n * (\s - 1)+\margin}:{360/\n * (\s)-\margin}:\radius); + } + \node[draw, circle, fill=green] at ({360/\n * (1 - 1)}:\radius) {}; + \node[draw, circle, fill=blue] at ({360/\n * (5 - 1)}:\radius) {}; + \node[draw, circle, fill=red] at ({360/\n * (9 - 1)}:\radius) {}; + \end{tikzpicture} + \end{figure} + + For three elements $a, b, c$ we say that $a\le_bc$ if after ''cutting'' the + cycle at $b$ we have a path from $a$ to $c$. + In this particular example we can say that the green element is + red-element-smaller than the blue one. \begin{example} - The class of all finitely generated vector spaces $\cV$ is a Fraïssé class - with WHP and CAP. + The class $\cC$ of all finite cyclic orders is a Fraïssé class, but does + not have WHP or CAP. \end{example} + + It is not hard to show that $\cC$ is indeed the Fraïssé class. As usual, + the hardest part is showing AP, which in this case is done analogously to + the linear orders. The Fraïssé limit of $\cC$ is a countable unit circle. + + $\cC$ hasn't WHP by the similar argument to this for linear + orderings. Imagine a cycling order of three elements and a partial automoprhism + with one fixed point and moving second element to the third. This cannot be + extended to automoprhism of any finite cyclic order. + + Also, $\cC$ cannot have CAP. A reason to that is that it do not admit + canonical amalgamation over the empty structure see this by taking + 1-element cyclic order and 3-element cyclic order with automoprhism other + than identity). + + % \begin{example} + % The class of all finitely generated vector spaces over a countable field + % $\cV$ 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 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. + \begin{example} - The class of all finite graphs $\cG$ is a + The class of all finite graphs $\cG$ is a Fraïssé class with WHP and free + amalgamation. \end{example} + + We have already shown this fact. Thus get that the random graph has a generic + automoprhism. + \begin{example} Graphs without triangles. \end{example} |