The normal hull of a completely simple semigroup

Abstract

A completely simple subsemigroup K of a completely simple semigroup S is a normal subsemigroup of S, or S is a normal extension of K, if x −1 Kx ⊆ K ( x ϵ S). A normal extension S of K is an essential extension if each non-trivial congruence on S restricts to a non-trivial congruence on K. For any completely simple semigroup S, the union Φ( S) of the automorphism groups of the maximal subgroups of S is endowed with a semigroup structure such that the mapping θ s of each element of S to the associated inner automorphism of the maximal subgroup containing it is a homomorphism of S into φ( S). It is shown that S has a maximal essential extension if and only if the metacentre of S (that is, the union of the centres of the maximal subgroups) is simply the set of idempotents of S. When such a maximal essential extension T exists, θ s is one-to-one and there exists a monomorphism of T onto Φ( S) extending θ s . A related semigroup ∑( S) whose elements are transformations of S with certain special properties (such as H -class preserving, isomorphisms on ℘-classes) is introduced and studied. A homomorphism of S into the product of ∑( S) and its left-right dual is constructed which induces the same congruence on S as θ s .