Preorder

A binary relation <= over a set X is a preorder if it is

1. reflexive, that is, for all a in X it holds that a <= a, and
2. transitive, that is, for all a, b and c in X it holds that if a <= b and b <= c the a < c.

If a preorder is also antisymmetric, that is, for all a and b in X it holds that if a <= b and b <= a then a = b, then it is a partial order.

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.