ℛ-, ℒ-, ℋ-commutative semigroups

Abstract

In this chapter we deal with semigroups in which the Green equivalence R (L, H) is a congruence. These semigroups are called R-commutative (L-commutative, H-commutative) semigroups. It is clear that a semigroup is H-commutative if and only if it is RR-commutative and L-commutative. We show that every R-commutative semigroup is a semilattice of archimedean semigroups. We note that, in general, the archimedean components are not RR-commutative. At the end of the chapter we deal with left soluble (right soluble, soluble) semigroups of length n. A monoid, with the identity e, is called soluble (right soluble, left soluble) of length n if it is H-commutative (R-commutative, L-commutative) and its n ^th derived (right derived, left derived) semigroup equals e. We show that a cancellative semigroup is soluble of length n if and only if it is both right and left soluble of length n. Moreover, a cancellative soluble semigroup of length n can be embedded in a soluble group of length n.