preorder
Definition
A preorder 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 preorder
Given such a relation, define a new relation $s\sim t$ on $S$ by
$$s\sim t\text{if and only if}s\lesssim t\text{and}t\lesssim s.$$ 
Then $\sim $ is an equivalence relation^{} on $S$, and $\lesssim $ induces a partial order^{} $\le $ on the set $S/\sim $ of equivalence classes^{} of $\sim $ defined by
$$[s]\le [t]\text{if and only if}s\lesssim t,$$ 
where $[s]$ and $[t]$ denote the equivalence classes of $s$ and $t$. In particular, $\le $ does satisfy antisymmetry, whereas $\lesssim $ may not.
Preorders as categories
A preorder $\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  preorder 
Canonical name  Preorder 
Date of creation  20130322 13:05:06 
Last modified on  20130322 13:05:06 
Owner  yark (2760) 
Last modified by  yark (2760) 
Numerical id  17 
Author  yark (2760) 
Entry type  Definition 
Classification  msc 06A99 
Synonym  preordering 
Synonym  preorder 
Synonym  preordering 
Synonym  quasiorder 
Synonym  quasiordering 
Synonym  quasiorder 
Synonym  quasiordering 
Synonym  semiorder 
Synonym  semiordering 
Synonym  semiorder 
Synonym  semiordering 
Related topic  WellQuasiOrdering 
Related topic  PartialOrder 
Defines  preordered 
Defines  preordered 
Defines  semiordered 
Defines  semiordered 
Defines  quasiordered 
Defines  quasiordered 