PreorderA binary relation <= over a set X is a preorder if it is
- reflexive, that is, for all a in X it holds that a <= a, and
- transitive, that is, for all a, b and c in X it holds that if a <= b and b <= c then a <= c.
A partial order can be constructed from a preorder by defining an equivalence relation
over X such that a b iff a <= b and b <= a. The relation implied by <= over the quotient set X /
, that is, the set of all equivalence classes defined by, then forms a partial order.