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\import{relation.tex}
\import{relation/properties.tex}
\subsection{Quasiorders}
% also called preorder
\begin{abbreviation}\label{quasiorder}
$R$ is a quasiorder iff
$R$ is quasireflexive and transitive.
\end{abbreviation}
% also called preorder
\begin{abbreviation}\label{quasiorder_on}
$R$ is a quasiorder on $A$ iff
$R$ is a binary relation on $A$ and
$R$ is reflexive on $A$ and transitive.
\end{abbreviation}
\begin{struct}\label{quasiordered_set}
A quasiordered set $X$ is a onesorted structure
equipped with
\begin{enumerate}
\item $\lt$
\end{enumerate}
such that
\begin{enumerate}
\item\label{quasiorder_type} $\lt[X]$ is a binary relation on $\carrier[X]$.
\item\label{quasiorder_refl} $\lt[X]$ is reflexive on $\carrier[X]$.
\item\label{quasiorder_tran} $\lt[X]$ is transitive.
\end{enumerate}
\end{struct}
\begin{lemma}\label{quasiorder_transitive_double}
Let $X$ be a quasiordered set.
Let $a, b, c, d \in X$.
Suppose $a\mathrel{\lt[X]} b\mathrel{\lt[X]} c\mathrel{\lt[X]} d$.
Then $a\mathrel{\lt[X]} d$.
\end{lemma}
\begin{proof}
$\lt[X]$ is transitive.
Thus $a\mathrel{\lt[X]} c\mathrel{\lt[X]} d$ by \hyperref[transitive]{transitivity}.
Hence $a\mathrel{\lt[X]} d$ by \hyperref[transitive]{transitivity}.
\end{proof}
\begin{proposition}\label{subseteqrel_is_quasiorder}
$\subseteqrel{A}$ is a quasiorder on $A$.
\end{proposition}
\begin{proof}
$\subseteqrel{A}$ is reflexive on $A$.
$\subseteqrel{A}$ is transitive.
\end{proof}
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