## Exhibit a semigroup which has a minimal but no least proper congruence

Exhibit a semigroup $S$ which has a minimal proper congruence, but no least proper congruence.

Let $S = \{a,b,0\}$ be a three-element zero semigroup. As we showed in this previous exercise, every equivalence relation on $S$ is a congruence.

The equivalences $\varepsilon_1 = \{\{a,0\},\{b\}\}$ and $\varepsilon_2 = \{\{b,0\},\{a\}\}$ (abusing notation; we give the classes rather than the pairs) are certainly minimal proper congruences, but $\varepsilon_1 \cap \varepsilon_2 = \Delta$. So $S$ has no least proper congruence.

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