Externally commutative semigroups

Abstract

In this chapter we deal with semigroups satisfying the identity axb = bxa. These semigroups are called externally commutative semigroups. It is clear that an externally commutative semigroup is medial. Thus the externally commutative semigroups are semilattice of externally commutative archimedean semigroups. A semigroup is externally commutative and 0-simple if and only if it is a commutative group with a zero adjoined. A semigroup is externally commutative and archimedean containing at least one idempotent element if and only if it is an ideal extension of a commutative group by an externally commutative nil semigroup. Moreover, every externally commutative archimedean semigroup without idempotent has a non-trivial group homomorphic image. We show that an externally commutative semigroup is regular if and only if it is a semilattice of commutative groups. We construct the least separative, left separative, right separative and weakly separative congruence on an externally commutative semigroup, respectively. We determine the subdirectly irreducible externally commutative semigroups. We prove that a semigroup is subdirectly irreducible and externally commutative with a globally idempotent core if and only if it is isomorphic to either G or G 0 or F, where G is a non-trivial subgroup of a quasicyclic p-group (p is a prime) and F is a two-element sernilattice. An externally commutative semigroup with a zero and a non-trivial annihilator is subbdirectly irreduucible if and only if it has a non-zero disjunctive element.