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partial order

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A \emph{partial order} (often simply referred to as an \emph{order} or \emph{ordering}) is a relation $\leq\:\subset A\times A$ that satisfies the following three properties:

\begin{enumerate}
    \item Reflexivity: $a \leq a$ for all $a\in A$
    \item Antisymmetry: If $a \leq b$ and $b \leq a$ for any $a, b\in A$, then $a = b$
    \item Transitivity: If $a \leq b$ and $b \leq c$ for any $a, b, c\in A$, then $a \leq c$
\end{enumerate}

A \emph{total order} is a partial order that satisfies a fourth property known as \emph{comparability}:

\begin{itemize}
\item Comparability:  For any $a,b\in A$, either $a\leq b$ or $b\leq a$.
\end{itemize}

A set and a partial order on that set define a poset.

\textbf{Remark}.  In some literature, especially those dealing with the foundations of mathematics, a partial order $\le$ is defined as a transitive irreflexive binary relation (on a set).  As a result, if $a\le b$, then $b \nleq a$, and therefore $\le$ is antisymmetric.
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