On finite simple semigroups

Abstract

Implicit operations (new operations commuting with all old homomorphisms) on pseudovarieties have been shown to play an important role in the study of these classes. They may be used to axiomatize sub-pseudovariaties and to describe recognizable subsets of (relatively) free objects. This paper presents a case study for the pseudovariety CS consisting of all finite simple semigroups. Based on a result of profinite group theory, a structural description of semigroups of implicit operations on finite simple semigroups is used to deduce that CS is join-irreducible.