On Embeddability of Idempotent Separating Extensions of Inverse Semigroups

E-solid locally inverse semigroups

“…Note that, when restricting our attention to inverse semigroups, the extensions considered in Theorem 3.1 are just the idempotent separating extensions. Thus the following weaker version of the main result of [8] easily follows from Theorem 3.1. Corollary 3.3.…”

“…The main result of [8] proves that, given a group variety U, if S is an inverse semigroup and ρ an idempotent separating congruence on S such that the idempotent classes of ρ belong to U then the extension (S, ρ) is embeddable in a λ-semidirect product extension of a member of U by S/ρ. The question naturally arises whether Theorem 3.1 can be strengthened so that the variety of all completely simple semigroups be replaced by any variety of completely simple semigroups.…”

Presentations of Inverse Semigroups, Their Kernels and Extensions

“…Of course, when S is a group this puts us exactly in the situation discussed above. Extensions of inverse semigroups are an important and well-studied area in the subject, with recent examples including [9,23]. The vast majority of research in this area has been concerned with proving structural results for certain special kinds of extension.…”

E-solid locally inverse semigroups as extensions

“…Note that, when restricting our attention to inverse semigroups, the extensions considered in Theorem 4.1.1 are just the idempotent separating extensions. Thus the following weaker version of the main result of [9] easily follows from Theorem 4.1.1.…”

“…Billhardt and Szittyai [9] strengthened the former result on idempotent separating extensions by proving that if S is an inverse semigroup and is an idempotent separating congruence such that every idempotent -class is from a group variety V then S is embeddable in a λ-semidirect product of a group from V by S/ .…”