In a zero semigroup, every equivalence relation is a congruence

Show that if S 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 S is a congruence.


Suppose S is a zero semigroup (with zero element 0). Let \sigma be an equivalence relation on S. Now 0 \sigma 0 since \sigma is reflexive. If a \sigma b and c \sigma d, then ac = 0 \sigma 0 = bd. Using this previous exercise, \sigma is a congruence on S.

Suppose S = \{a,b\} is a semilattice of order 2. Now a^2 = a and b^2 = b, and ab = ba. Without loss of generality, we can say ab = b, since otherwise we can simply switch the roles of a and b. Now S has only two equivalence relations, \Delta and S^2, which are congruences on any semigroup.

Suppose S is a left zero semigroup and \sigma an equivalence on S. (So ab = a for all a,b \in S.) If a \sigma b and c \sigma d, then ac = a \sigma b = bd. So \sigma is a congruence. Likewise, if S is a right zero semigroup, then every equivalence is a congruence.

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