$E$-solid locally inverse semigroups

TL;DR

This paper characterizes $E$-solid locally inverse semigroups by showing they can be embedded into $bbb4$-semidirect products of completely simple semigroups and inverse semigroups, revealing their structural properties.

Contribution

It establishes a new embedding theorem for $E$-solid locally inverse semigroups into $bbb4$-semidirect products, linking their structure to completely simple and inverse semigroups.

Findings

$E$-solid locally inverse semigroups embed into $bbb4$-semidirect products.

They are regular subsemigroups of such $bbb4$-semidirect products.

The structure relates to completely simple and inverse semigroups.

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.