Commutators in completely simple semigroups

We obtain a characterization of the binary commutator on completely simple semigroups, using their Rees matrix representation. Consequently, we prove that a regular semigroup is left or right nilpotent (solvable) if and only if it is simple, and all its \(\mathscr {H}\) $\mathcal{H}$ -classes are nilpotent (solvable) groups.