Regular congruences on an idempotent-regular-surjective semigroup

A semigroup $S$ is called \emph{idempotent-surjective} [\emph{regular-surjective}] if whenever $\rho$ is a congruence on $S$ and $a\rho$ is idempotent [regular] in $S/\rho$, then there is $e\in E_S\cap a\rho$ [$r\in Reg(S)\cap a\rho$], where $E_S$ [$Reg(S)$] denotes the set of idempotents [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] semigroup is uniquely determined by its kernel and trace [the set of equivalence classes containing idempotents]. Finally, we prove that all structurally regular semigroups are idempotent-regular-surjective. 10.1017/S000497271300027