WE-m semigroups

Abstract

In this chapter we deal with semigroups in which, for 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 , where m is a fixed i n te g er m ≥ 2. These se m igrou p s are c a lled WE- m se m igroups. It is clear that every E-m semigroup is a WE-m semigroup. The examination of WE-m semigroups need some results about E-m semigroups. Thus the E-m semigroups were examined in the previous chapter. As a WE-m semigroup is a left and right Putcha semigroup, it is a semilattice of WE-m archimedean semigroups. We show that the 0-simple WE-mn semigroups are the completely simple E-m semigroups with a zero adjoined. A semigroup is a WE-m archimedean 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 semigroup. We also prove that every WE-2 archimnedean semigroup without idempotent element has a non-trivial group homomorphic image. We deal with the regular WE-m semigroups. We show that the regular WE-m semigroups are exactly the regular exponential semigroups. Moreover, we show that a semigroup which is an ideal extension of a regular semigroup K by a nil sernigroup N is a WE-2 semigroup if and only if K is an E-2 semigroup and the extension is retract. We deal with the subdirectly irreducible WE-2 semigroups.