The set of P-partial kernel normal systems for a semigroup S with   non-empty set of idempotents forms a complete lattice, which is  a completely -homomorphic image of  Every strongly regular congruence on an E-inversive semigroup is uniquely determined  by its P-partial kernel normal system. These results generalize  the corresponding results for eventually regular semigroups and  E-inversive semigroups.

semigroup with non-empty set of idempotents, P-partial kernel normal system, kernel normal system, lattice of congruences, E-inversive semigroup, strongly regular congruence.