Abstract. We prove that if $S$ is an $E$-solid locally inverse semigroup, and $\rho $ is an inverse semigroup congruence on $S$ such that the idempotent classes of $\rho $ are completely simple semigroups then $S$ is embeddable into a $\lambda $-semidirect product of a completely simple semigroup by $S/\rho $. Consequently, the $E$-solid locally inverse semigroups turn out to be, up to isomorphism, the regular subsemigroups of $\lambda $-semidirect products of completely simple semigroups by inverse semigroups.
AMS Subject Classification
(1991): 20M10, 20M17
E-solid locally inverse semigroups,
$\lambda $-semidirect product,
Received July 3, 2018. (Registered under 61/2018.)