Rees Matrix Covers and Semidirect Products of Regular Semigroups
Journal of Algebra
In a recent paper (Trans. Amer. Math. Soc. 349 (1997), 4265–4310), P.G. Trotter and the author introduced a “regular” semidirect product U ∗ V of e-varieties U and V. Among several specific 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 U∗RZ in a number of ways; one in terms of an associated category SE 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 than “e-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 verified. 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. K. Auinger and P. G. Trotter (Pseudovarieties, regular semigroups and semidirect products, J. London Math. Soc. (2)58(1998), 284–296) have used our results to describe the pseudovarieties generated by several important classes of (finite) regular semigroups.