E-m semigroups, exponential semigroups
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.
KeywordsAbelian Group Inverse Semigroup Regular Semigroup Simple Semigroup Idempotent Element
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