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Preorder

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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 then 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.



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