## 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.