REGULAR CONGRUENCES ON AN IDEMPOTENT-REGULAR-SURJECTIVE SEMIGROUP

A semigroup $S$ is called idempotent-surjective (respectively, regular-surjective) if whenever $\rho $ is a congruence on $S$ and $a\rho $ is idempotent (respectively, regular) in $S/ \rho $, then there is $e\in {E}_{S} \cap a\rho $ (respectively, $r\in \mathrm{Reg} (S)\cap a\rho $), where ${E}_{S} $ (respectively, $\mathrm{Reg} (S)$) denotes the set of all idempotents (respectively, regular elements) of $S$. Moreover, a semigroup $S$ is said to be idempotent-regular-surjective if it is both idempotent-surjective and regular-surjective. We show that any regular congruence on an idempotent-regular-surjective (respectively, regular-surjective) semigroup is uniquely determined by its kernel and trace (respectively, the set of equivalence classes containing idempotents). Finally, we prove that all structurally regular semigroups are idempotent-regular-surjective.