Given a semigroup and a congruence on , show that the mapping given by is a surjective semigroup homomorphism.
Let be a homomorphism of semigroups. Define a relation by if and only if . Show that is a congruence on . Show further that .
Recall that is a semigroup under the operator , which is well-defined (i.e. does not depend on the representatives chosen) since is a congruence. Now , so that is a semigroup homomorphism. Moreover, we have for all , so that is surjective.
Now let and be as described above. Certainly is an equivalence. Now if and , then and , so that . Hence , and so is a congruence on .
Now define by . Certianly is total, and we claim it is well defined. Indeed, if we have such that , then . So in fact is a function. Since , is a semigroup homomorphism. If , then we have , so that . Hence is injective. Finally, if , with , then . So is onto the subset , and we have .