# pre-order

## 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).

 Title pre-order Canonical name Preorder Date of creation 2013-03-22 13:05:06 Last modified on 2013-03-22 13:05:06 Owner yark (2760) Last modified by yark (2760) Numerical id 17 Author yark (2760) Entry type Definition Classification msc 06A99 Synonym pre-ordering Synonym preorder Synonym preordering Synonym quasi-order Synonym quasi-ordering Synonym quasiorder Synonym quasiordering Synonym semi-order Synonym semi-ordering Synonym semiorder Synonym semiordering Related topic WellQuasiOrdering Related topic PartialOrder Defines pre-ordered Defines preordered Defines semi-ordered Defines semiordered Defines quasi-ordered Defines quasiordered