Let be a semigroup and let denote the set of idempotents in . Define an order on by iff . Show that is a partial order.
Recall that a partial order is a binary relation which is reflexive, antisymmetric, and transitive.
If , then is idempotent, so that . So .
Suppose such that and . Then and , so that .
Suppose such that and . Now and . So we have and , and thus .