Comparing the regular and the restricted regular semidirect products

Abstract

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 S * r V S * rr (Aq • V)are proved where S is the e-variety of all semilattices and Aq 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.

Rees matrix semigroups and the regular semidirect product

“…Let 5 and T be semigroups. If 5 is regular and T is a homomorphic image of a [3] Rees matrix semigroups 41 regular subsemigroup of 5 then we say that T regularly divides S. We use Reg(S) to denote the set of all regular elements in S and End 5 the endomorphism monoid of 5. Let / and A be non-empty sets.…”

“…The results of this paper have been obtained while investigating how far the restricted regular semidirect product and the regular semidirect product of the variety of semilattices by a variety of completely simple semigroups can be from each other. The results in this direction are published in [3].…”