Weakly exponential semigroups

Abstract

In the previous chapter we dealt with semigroups in which, for a fixed integer m ≥ 2 and every elements a and b, there is a non-negative integer k such that (ab) _m+k =a _m b _m =(ab) _k a _m b _m . In this chapter we deal with semigroups which satisfy this condition for every integer m ≥ 2. These semigroups are called weakly exponential semigroups. It follows from results of the previous chapter that every weakly exponential semigroup is a semilattice of weakly exponential archimedean semigroups. A semigroup is a weakly exponential archimedean semigroup containing at least one idempotent element if and only if it is a retract extension of a rectangular abelian group by a nil semigroup. It is also proved that every weakly exponential archimedean semigroup without idempotent element has a non-trivial group homomorphic image. We prove that every weakly exponential semigroup is a band of weakly exponential t-archimedean semigroups. As a consequence of the previous chapter, a semigroup is a subdirectly irreducible weakly exponential semigroup with a globally idempotent core if and only if it is isomorphic to either G or G ⁰ or B, where G is a non-trivial subgroup of a quasicyclic p-group (p is a prime) and B is a non-trivial subdirectly irreducible band. At the end of the chapter, we determine the weakly exponential ∆-semigroups. We prove that a semigroup S is a weakly exponential ∆-semigroup if and only if one of the following satisfied. (1) S is isomorphic to either G or G ⁰, where G is a non-trivial subgroup of a quasicyclic p-group.