# Preorder

In mathematics, especially in order theory, preorders are certain kinds of binary relations that are closely related to partially ordered sets. The name quasiorder is also a common expression for preorders. Many order theoretical definitions for partially ordered sets can be generalized to preorders, but the extra effort of generalization is rarely needed.

## Formal definition

Consider some set P and a binary relation ≤ on P. Then ≤ is a preorder, or quasiorder, if it is reflexive and transitive, i.e., for all a, b and c in P, we have that:

aa (reflexivity)
if ab and bc then ac (transitivity)

A set that is equipped with a preorder is called a preordered set. If a preorder is also antisymmetric, that is, ab and ba implies a = b, then it is a partial order.

A partial order can be constructed from any preorder by identifying "equal" points. Formally, one defines an equivalence relation ~ over X such that a ~ b iff ab and ba. Now the quotient set X / ~, i.e. the set of all equivalence classes of ~, can easily be ordered by defining [x] ≤ [y] iff xy. By the construction of ~ this definition is independent from the chosen representatives and the corresponding relation is indeed well-defined. It is readily verified that this yields a partially ordered set.

## Examples of preorders

• Art and Cultures
• Countries of the World (http://www.academickids.com/encyclopedia/index.php/Countries)
• Space and Astronomy