Preorder
A 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.
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.
- See also : Mathematics
Referenced By
List of order topics | Order theory glossary
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