Right commutative semigroups

Abstract

In this chapter we deal with semigroups which satisfy the identity axy = ayx. These semigroups are called right commutative semigroups. It is clear that a right commutative semigroup is medial and so we can use the results of the previous chapter for right commutative semigroups. For example, every right commutative semigroup is a semilattice of right commutative archimedean semigroups and is a band of right commutative t-archimedean semigroups. A semigroup is right commutative and simple if and only if it is a left abelian group. Moreover, a semigroup is right commutative and archimedean containing at least one idempotent element if and only if it is a retract extension of a left abelian group by a right commutative nil semigroup. We characterize the right commutative left cancellative and the right commutative right cancellative semigroups, respectively. Clearly, a semigroup is right commutative and left cancellative if and only if it is a commutative cancellative semigroup. A semigroup is right commutative and right cancellative if and only if it is embeddable into a left abelian group if and only if it is a left zero semigroup of commutative cancellative semigroups. It is shown that a right commutative semigroup is embeddable into a semigroup which is a union of groups if and only if it is right separative.