A (reflexive) preorder relation on set is a binary relation that is reflexive and transitive, that is, these two properties hold:
Constructing a partial order from a preorder
Intuitively, preorder differs from partial order in that there are distinct elements that have same ordering properties with respect to other elements. For such elements we therefore have and . By identifying these elements we obtain a partial order.
More precisely, let be a preorder. Define relation by
It is easy to verify that is an equivalence relation. Moreover, if we define relation on equivalence classes by
for , then we can prove that is a partial order.