Definition

A pre-order on a set $S$ is a relation $\lesssim$ on $S$ satisfying the following two axioms:

reflexivity: $s \lesssim s$ for all $s \in S$ , and

transitivity: If $s \lesssim t$ and $t \lesssim u$ , then $s \lesssim u$ ; for all $s,t,u \in S$ .

Partial order induced by a pre-order

Given such a relation, define a new relation $s\sim t$ on $S$ by$$ s\sim t \hbox{ if and only if } s\lesssim t \hbox{ and } t \lesssim s.$$ Then $\sim$ 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$$ [s] \leq [t] \hbox{ if and only if } s \lesssim t,$$ where $[s]$ and $[t]$ denote the equivalence classes of $s$ and $t$ . In particular, $\leq$ does satisfy antisymmetry, whereas $\lesssim$ may not.

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).