| Version 9 |
Version 8 |
| section*{Definition} |
|
|
|
| A \emph{pre-order} on a set $S$ is a relation $\lesssim$ on $S$ satisfying the following two axioms: |
A \emph{pre-order} on a set $S$ is a relation $\lesssim$ on $S$ satisfying the following two axioms: |
| \begin{description} |
\begin{description} |
| \item reflexivity: $s \lesssim s$ for all $s \in S$, and |
\item reflexivity: $s \lesssim s$ for all $s \in S$, and |
| \item transitivity: If $s \lesssim t$ and $t \lesssim u$, then $s \lesssim u$; for all $s,t,u \in S$. |
\item transitivity: If $s \lesssim t$ and $t \lesssim u$, then $s \lesssim u$; for all $s,t,u \in S$. |
| \end{description} |
\end{description} |
|
|
| \section*{Partial order induced by a pre-order} |
|
|
|
| Given such a relation, the relation $s\sim t:\Leftrightarrow (s\lesssim t) \wedge (t \lesssim s)$ is an equivalence relation on $S$, and $\lesssim$ induces a partial order $\leq$ on the set $S/\sim$ of equivalence classes of $\sim$ defined by |
Given such a relation, the relation $s\sim t:\Leftrightarrow (s\lesssim t) \wedge (t \lesssim s)$ is an equivalence relation on $S$, and $\lesssim$ induces a partial order $\leq$ on the set $S/\sim$ of equivalence classes of $\sim$ defined by |
| \[ [s]_\sim \leq [t]_\sim :\Leftrightarrow s \lesssim t,\] |
\[ [s]_\sim \leq [t]_\sim :\Leftrightarrow s \lesssim t,\] |
| where $[s]_\sim$ and $[t]_\sim$ denote the equivalence classes of $s$ and $t$. In particular, $\leq$ does satisfy antisymmetry, whereas $\lesssim$ may not. |
where $[s]_\sim$ and $[t]_\sim$ denote the equivalence classes of $s$ and $t$. In particular, $\leq$ does satisfy antisymmetry, whereas $\lesssim$ may not. |
|
|
| \section*{Pre-orders as categories} |
|
|
|
| A pre-order $\lesssim$ on a set $S$ can be considered as a small category, in the which the objects are the elements of $S$ and there is a unique morphism from $x$ to $y$ if $x\lesssim y$ (and none otherwise). |
|
