Mathematical Structures: Directed graphs

# Directed graphs

Difference (from prior major revision) (no other diffs)

Changed: 28,30c28,29
 \Large Name of class \quad\href{http://math.chapman.edu/cgi-bin/structures?action=edit;id=Template}{edit} % Note: replace "Template" with Name_of_class in previous line

Changed: 32c31
 \abbreviation{Abbr}
 \abbreviation{DiGraph}

Changed: 35,40c34
 A ... is a structure $\mathbf{A}=\langle A,...\rangle$ of type $\langle ...\rangle$ such that $\langle A,...\rangle$ is a \href{Name_of_class.pdf}{name of class} $op_1$ is (name of property): $axiom_1$
 A directed graph (or digraph for short) is a structure $\mathbf{G}=\langle G,E\rangle$ such that

Changed: 42c36
 $op_2$ is ...: $...$
 $E$ is binary relation on $G$: $E\subseteq G\times G$

Changed: 51,52c45,46
 Let $\mathbf{A}$ and $\mathbf{B}$ be ... . A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a homomorphism: $h(x ... y)=h(x) ... h(y)$
 Let $\mathbf{G}$ and $\mathbf{H}$ be directed graphs. A morphism from $\mathbf{G}$ to $\mathbf{H}$ is a function $h:G\rightarrow H$ that preserves $E$: $\langle x,y\rangle\in E^{\mathbf G}\implies \langle h(x), h(y)\rangle\in E^{\mathbf H}$

Changed: 78,81c72,75
 Classtype & (value, see description) \cite{Ln19xx} \\\hline Equational theory & \\\hline Quasiequational theory & \\\hline First-order theory & \\\hline
 Classtype & variety \\\hline Equational theory & decidable\\\hline Quasiequational theory & decidable\\\hline First-order theory & undecidable\\\hline

Changed: 84,88c78,82
 Congruence distributive & \\\hline Congruence modular & \\\hline Congruence $n$-permutable & \\\hline Congruence regular & \\\hline Congruence uniform & \\\hline
 Congruence distributive & no\\\hline Congruence modular & no\\\hline Congruence $n$-permutable & no\\\hline Congruence regular & no\\\hline Congruence uniform & no\\\hline

Changed: 92,93c86,87
 Amalgamation property & \\\hline Strong amalgamation property & \\\hline
 Amalgamation property & yes\\\hline Strong amalgamation property & yes\\\hline

http://mathcs.chapman.edu/structuresold/files/Directed_graphs.pdf
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\newtheorem{definition}{Definition}
\newtheorem*{morphisms}{Morphisms}
\newtheorem*{basic_results}{Basic Results}
\newtheorem*{examples}{Examples}
\newtheorem{example}{}
\newtheorem*{properties}{Properties}
\newtheorem*{finite_members}{Finite Members}
\newtheorem*{subclasses}{Subclasses}
\newtheorem*{superclasses}{Superclasses}
\newcommand{\abbreviation}[1]{\textbf{Abbreviation: #1}}
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\begin{document}
\textbf{\Large Directed graphs}

\abbreviation{DiGraph}

\begin{definition}
A \emph{directed graph} (or \emph{digraph} for short) is a structure $\mathbf{G}=\langle G,E\rangle$ such that

$E$ is binary relation on $G$:  $E\subseteq G\times G$

Remark: This is a template.

It is not unusual to give several (equivalent) definitions. Ideally, one of the definitions would give an irredundant axiomatization that does not refer to other classes.
\end{definition}

\begin{morphisms}
Let $\mathbf{G}$ and $\mathbf{H}$ be directed graphs. A morphism from $\mathbf{G}$ to $\mathbf{H}$ is a function $h:G\rightarrow H$ that preserves $E$:
$\langle x,y\rangle\in E^{\mathbf G}\implies \langle h(x), h(y)\rangle\in E^{\mathbf H}$
\end{morphisms}

\begin{definition}
An \emph{...} is a structure $\mathbf{A}=\langle A,...\rangle$ of type $\langle ...\rangle$ such that

$...$ is ...:  $axiom$

$...$ is ...:  $axiom$
\end{definition}

\begin{basic_results}
\end{basic_results}

\begin{examples}
\begin{example}
\end{example}
\end{examples}

\begin{table}[h]
\begin{properties} (\href{http://math.chapman.edu/cgi-bin/structures?Properties}{description})

Feel free to add or delete properties from this list. The list below may contain properties that are not relevant to the class that is being described.

\begin{tabular}{|ll|}\hline
Classtype                       & variety \\\hline
Equational theory               & decidable\\\hline
Quasiequational theory          & decidable\\\hline
First-order theory              & undecidable\\\hline
Locally finite                  & \\\hline
Residual size                   & \\\hline
Congruence distributive         & no\\\hline
Congruence modular              & no\\\hline
Congruence $n$-permutable       & no\\\hline
Congruence regular              & no\\\hline
Congruence uniform              & no\\\hline
Congruence extension property   & \\\hline
Definable principal congruences & \\\hline
Equationally def. pr. cong.     & \\\hline
Amalgamation property           & yes\\\hline
Strong amalgamation property    & yes\\\hline
Epimorphisms are surjective     & \\\hline
\end{tabular}
\end{properties}
\end{table}

\begin{finite_members} $f(n)=$ number of members of size $n$.

$\begin{array}{lr} f(1)= &1\\ f(2)= &\\ f(3)= &\\ f(4)= &\\ f(5)= &\\ \end{array}$\qquad
$\begin{array}{lr} f(6)= &\\ f(7)= &\\ f(8)= &\\ f(9)= &\\ f(10)= &\\ \end{array}$

\end{finite_members}

\begin{subclasses}\

\href{....pdf}{...} subvariety

\href{....pdf}{...} expansion

\end{subclasses}

\begin{superclasses}\

\href{....pdf}{...} supervariety

\href{....pdf}{...} subreduct

\end{superclasses}

\begin{thebibliography}{10}

\bibitem{Ln19xx}
F. Lastname, \emph{Title}, Journal, \textbf{1}, 23--45 \href{http://www.ams.org/mathscinet-getitem?mr=12a:08034}{MRreview}

\end{thebibliography}

\end{document}
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