Regular WE-2 semigroups (Q1084515)

Let m be a positive integer greater than one. A semigroup is called an E- m semigroup if it satisfies the identity \((xy)^ m=x^ my^ m\). A WE-m semigroup is a semigroup S such that for every x,y\(\in S\), there exists a positive integer k such that \((xy)^{m+k}=x^ my^ m(xy)^ k=(xy)^ kx^ my^ m\). It is shown that the regular WE-2 semigroups are exactly the regular E-2 semigroups and that an ideal extension of a regular semigroup K by a nil semigroup N is a WE-2 semigroup iff K is an E-2 semigroup and the extension is a retract extension.