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