Rees Matrix Covers and Semidirect Products of Regular Semigroups

In a recent paper, P.G. Trotter and the author introduced a \regular" semidirect product UV of e-varieties U and V. Among several speciic situations investigated there was the case V = RZ, the e-variety of right zero semigroups. Applying a covering theorem of McAlister, it was shown there that in several important cases (for instance for the e-variety of inverse semigroups), U RZ is precisely the e-variety LU of \locally-U" semigroups. The main result of the current paper characterizes membership of a regular semigroup S in URZ in a number of ways, one in terms of an associated category S E and another in terms of S regularly dividing a regular Rees matrix semigroup over a member of U. The categorical condition leads directly to a characterization of the equality U RZ = LU in terms of a graphical condition on U, slightly weaker thanè-locality'. Among consequences of known results on e-locality, the conjecture CR RZ = LCR, (with CR denoting the e-variety of completely regular semigroups), is therefore veriied. The connection with matrix semigroups then leads to a range of Rees matrix covering theorems that, while slightly weaker than McAlister's, apply to a broader range of examples. Auinger and Trotter have used our results to describe the pseudovarieties generated by several important classes of ((nite) regular semigroups. An e-variety of regular semigroups is a class of regular semigroups that is closed under products, quotients and regular subsemigroups. In a recent paper 6], P.G. Trotter and the author introduced a product U V of e-varieties U;V, well deened if (and only if) either U or V consists of completely simple semigroups, as follows: UV is the e-variety 1