Let be a semigroup and let be a congruence on . Let denote the set of all congruences on which contain , and let denote the set of all congruences on . Note that both and are partially ordered by . Show that in fact they are isomorphic as partially ordered sets.

Given , define a relation on by if and only if there exist and such that . Since , note that in fact if and only if .

We claim that is a congruence on . Since is reflextive, we have for all , and thus for all . If , then , so that , and thus . If and , then , so that and hence . Finally, if and , then and , so that , and hence . So is a congruence on .

Now define by . We claim that is a poset isomorphism.

Suppose . Now for all , we have if and only if if and only if if and only if . So . Thus is injective.

Now suppose . Define a relation on by if and only if . We claim that is a congruence. Indeed, for all , , so ; if then , so that , hence ; if and , then , so that , and thus ; and if and , then and , so , and thus . We claim also that . Indeed, if , then , so that and we have . So in fact . Finally, we claim that ; in fact, if and only if if and only if as desired. So is surjective.

So is a bijection. Finally, we claim that preserves set containment. Indeed, if , then implies , implies , implies . So .

Hence and are isomorphic as partially ordered sets.