Show that if is (1) a zero semigroup, (2) a semilattice of order 2, (3) a left zero semigroup, or (4) a right zero semigroup, then every equivalence relation on is a congruence.
Suppose is a zero semigroup (with zero element ). Let be an equivalence relation on . Now since is reflexive. If and , then . Using this previous exercise, is a congruence on .
Suppose is a semilattice of order 2. Now and , and . Without loss of generality, we can say , since otherwise we can simply switch the roles of and . Now has only two equivalence relations, and , which are congruences on any semigroup.
Suppose is a left zero semigroup and an equivalence on . (So for all .) If and , then . So is a congruence. Likewise, if is a right zero semigroup, then every equivalence is a congruence.