A partially ordered set is a set with a relation on satisfying:
(1) for all (reflexivity)
(2) if and , then (antisymmetry)
(3) if and , then (transitivity)
Note that for two given elements and , it may not be the case that and are comparable, that is, or . If this is true for all pairs and in , we say that is totally ordered.
The set of people in the world, with defined by "is a direct descendant of," is a partially ordered set. The set of positive integers, with defined by "divides," is a partially ordered set.
Note that neither of these sets are totally ordered. A brother and sister are not comparable in , since they are not direct descendants of each other; and and are not comparable in , since neither divides the other.