# E-m semigroups, exponential semigroups

• Attila Nagy
Chapter
Part of the Advances in Mathematics book series (ADMA, volume 1)

## Abstract

In this chapter we deal wih the E-m sem.igroups and the exponential semigroups. A semigroup is called an E-m semigroup (m is an integer with m ≥ 2) if it satisfies the identity (ab) m = a m b m . A semigroup which is an E-m semigroup for every integer m ≥ 2 is called an exponential semigroup. We show that a semigroup is an exponential semigroup if and only if it is an E-2 and E-3 semigroup. It is proved that every E-m semigroup (exponential semigroup) is a semilattice of archimedean E-m semigroups (exponential semigroups). It is also shown that every exponential semigroup is a band of t-archimedean semigroups. We show that a semigroup is a 0-simple E-m semigroup if and only if it is a completely simple E-m semigroup with a zero adjoined. We characterize the completely simple E-m semigroups and show that a semigroup is an archimedean E-m semigroup containing at least one idempotent element if and only if it is a retract extension of a completely simple E-m semigroup by a nil E-m semigroup. It is proved that every archimedean E-2 semigroup without idempotent has a non-trivial group homomorphic image. We show that a regular E-m semigroup is a semilattice of completely simple E-m semigroups. Moreover, a semigroup is an inverse E-m semigroup if and only if it is a semilattice of E-m groups. We deal with the regular E-2 semigroups. We show that a semigroup is a regular E-2 semigroup if and only if it is a spined product of some band and a semilattice of abelian groups and so it is a regular exponential semigroup. At the end of the chapter we describe the translational hull of a regular E-2 semigroup.

## Keywords

Abelian Group Inverse Semigroup Regular Semigroup Simple Semigroup Idempotent Element
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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## Copyright information

© Springer Science+Business Media Dordrecht 2001

## Authors and Affiliations

• Attila Nagy
• 1
1. 1.Department of Algebra, Institute of MathematicsBudapest University of Technology and EconomicsHungary