On the decomposition of completely regular ordered semigroups into union of left and right simple ordered semigroups

Abstract

We give sufficient conditions (Rs-condition and Ls-condition) which a completely regular ordered semigroup S must satisfy so that the set Ge={a∈S|a∈(eS]∩(Se],e∈(aS]∩(Sa]},e∈E(S), is (maximal under the inclusion relation) right and left simple subsemigroup of S, where E(S) is the set of elements of S for which e≤e². We prove that S=U_e∈E(S)Ge and thus S is decomposed into a union of (disjoint) right and left simple semigroups. In addition every N-class of a completely regular ordered semigroup satisfying Rs-condition and Ls-condition is a union of right and left simple subsemigroups of S, where N is the least complete semilattice congruence on S. Finally we prove that the previous decompositions into unions of right and left simple semigroups characterize equivalently an ordered semigroup satisfying both Rs-condition and Ls-condition in order to be a completely regular ordered semigroup.