Mathematical Structures: Preordered sets

# Preordered sets

Difference (from prior major revision) (author diff)

Changed: 30c30,31
 \abbreviation{}
 \abbreviation{Qoset}

Changed: 32,34c33,34
 A preordered set (also called a quasi-ordered set or qoset for short) is a structure $\mathbf{P}=\left\langle P,\preceq \right\rangle$ such that $P$ is a set and $\preceq$ is a binary relation on $P$ that is
 A preordered set (also called a quasi-ordered set or qoset for short) is a structure $\mathbf{P}=\langle P,\preceq\rangle$ such that $P$ is a set and $\preceq$ is a binary relation on $P$ that is

http://mathcs.chapman.edu/structuresold/files/Preordered_sets.pdf
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\newtheorem{definition}{Definition}
\newtheorem*{morphisms}{Morphisms}
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\begin{document}
\textbf{\Large Preordered sets}

\abbreviation{Qoset}

\begin{definition}
A \emph{preordered set} (also called a \emph{quasi-ordered set} or \emph{qoset} for short) is a structure $\mathbf{P}=\langle P,\preceq\rangle$
such that $P$ is a set and $\preceq$ is a binary relation on $P$ that is

reflexive:  $x\preceq x$ and

transitive:  $x\preceq y \text{ and } y\preceq z\implies x\preceq z$

Remark:
\end{definition}

\begin{morphisms}
Let $\mathbf{P}$ and $\mathbf{Q}$ be qosets. A morphism from $\mathbf{P}$ to
$\mathbf{Q}$ is a function $f:P\rightarrow Q$ that is preorder-preserving:

$x\preceq y\implies f(x)\preceq f(y)$
\end{morphisms}

\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})

\begin{tabular}{|ll|}\hline
Classtype & Universal Horn class\\\hline
Universal theory & Decidable\\\hline
First-order theory & Undecidable\\\hline
Amalgamation property & \\\hline
Strong amalgamation property & \\\hline
Epimorphisms are surjective & \\\hline
\end{tabular}
\end{properties}
\end{table}
\begin{finite_members} $f(n)=$ number of members of size $n$.

$\begin{array}{lr} f(1)= &1\\ f(2)= &2\\ f(3)= &\\ f(4)= &\\ f(5)= &\\ f(6)= &\\ f(7)= &\\ \end{array}$
\end{finite_members}
\hyperbaseurl{http://math.chapman.edu/structures/files/}
\parskip0pt
\begin{subclasses}\

\href{Posets.pdf}{Posets}

\href{Connected_qosets.pdf}{Connected qosets}

\end{subclasses}
\begin{superclasses}\

\href{Binary_relational_structures.pdf}{Binary relational structures}

\end{superclasses}

\begin{thebibliography}{10}

\bibitem{Ln19xx}

\end{thebibliography}

\end{document}
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