Essential normal and conjugate extensions of inverse semigroups

In the following we use the notation and terminology of [6] and [7]. If S is an inverse semigroup, then Es denotes the semilattice of idempotents of S. If a is any element of the inverse semigroup, then a−1 denotes the inverse of a in S. An inverse subsemigroup S of an inverse semigroup S′ is self-conjugate in S′ if for all x ∈ S′,x−1Sx ⊆ S; if this is the case, S′ is called a conjugate extension of S. An inverse subsemigroup S of S′ is said to be a full inverse subsemigroup of S′ if Es = Es′. If S is a full self-conjugate inverse subsemigroup of the inverse semigroup S′, then S is called a normal inverse subsemigroup of S′, or, S′ is called a normal extension of S.