Definition:Preordering

Definition
Let $S$ be a set.

Definition 1
A preordering on $S$ is a relation $\mathcal R$ on $S$ such that:

Definition 2
Symbols used to define such a general preordering relation are often variants on $\lesssim$, $\precsim$ or $\precapprox$.

A symbol for a preordering can be reversed, and the sense is likewise inverted:


 * $a \precsim b \iff b \succsim a$

Partial vs. Total Preorderings
Note that this definition of preordering does not demand that every pair of elements of $S$ is related by $\precsim$. The way we have defined a preordering, they may be, or they may not be, depending on the context.

If it is the case that $\precsim$ is a connected relation, i.e. that every pair of elements is related by $\precsim$, then $\precsim$ is called a total preordering.

If it is not the case that $\precsim$ is connected, then $\precsim$ is called a partial preordering.

Also known as
A preordering is also known as a preorder.

Either name can be seen with a hyphen: pre-ordering and pre-order.

Some sources use the term quasiorder or quasi-order.

uses the term reflexive partial ordering, but as this can so easily be confused with the concept of a partial ordering this term is not recommended.

Also see

 * Symmetric Preordering is Equivalence Relation
 * Antisymmetric Preordering is Ordering


 * Definition:Preorder Category, interpreting preorders as categories.