Comparing the regular and the restricted regular semidirect products

We compare the two recently introduced semidirect product operations (*r) and (*rr) within the lattice of e-varieties of locally inverse semigroups. For each e-variety V which contains all rectangular bands and is properly contained in the e-variety of all completely simple semigroups, the inclusions S (*rr) V subset of or equal to S-*r V subset of or equal to S (*rr) (Aq o V) are proved where S is the e-variety of all semilattices and A(q) the variety of all abelian groups of exponent dividing q where q is any integer greater than one. Some consequences for the class of finite locally inverse semigroups are also obtained.