∛C-commutative semigroups

Abstract

In this chapter we deal with semigroups which are both ∛-commutative and conditionally commutative. These semigroups are called ∛C-commutative semigroups. The ∛-commutative semigroups and the conditionally commutative semigroups are examined in Chapter 5 and Chapter 6, respectively. From the results of those chapters it follows that every C-commutative semigroup is a semilattice of conditionally commutative a∛Chimedean semigroups. In this chapter, we show that the simple ∛C-commutative semigroups are exactly the right abelian groups. By the help of this result we show that every ∛C-commutative a∛Chimedean semigroup containing at least one idempotent element is an ideal extension of a right abelian group by a commutative nil semigroup. As a consequence, we prove that every ∛C-commutative regular semigroup is a spined product of a right normal band and a semilattice of abelian groups. We determine the subdirectly irreducible ∛C-commutative semigroups with a globally idempotent core. We show that they are those semigroups which are isomorphic to either G or G0 or F or R or R0, where G is a non-trivial subgroup of a quasicyclic p-group (p is a prime), F is a two-element semilattice and R is a two-element right zero semigroup. At the end of the chapter we deal with the ∛C-commutative G0-semigroups. It is shown that a semigroup S is an Ccommutative d-semigroup if and only if it is isomorphic to either G or G0 or R or R0 or N or N 1 , where G is a non-trivial subgroup of a quasicyclic p-group (p is a prime), R is a two-element right zero semigroup and N is a commutative nil semigroup whose ideals form a chain with respect to inclusion.