Rees Matrix Covers and Semidirect Products of Regular Semigroups

Abstract

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 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 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.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], Trotter and the author introduced a product U * V of e-varieties U V, well-defined if (and only if) either U or V consists of completely simple semigroups, as follows: U * V is the e-variety generated by the semigroups of regular elements of the wreath products (or the semidirect prod-287

Rees matrix semigroups and the regular semidirect product

“…In particular, if T is a group then s, is an automorphism for every t e T. Note that, in [9], the condition on S\ is not required, and an action with this additional property is termed left unitary.…”

“…A semidirect or wreath product of regular semigroups need not be regular. However, a regular version of the semidirect product was introduced in [9] as follows. It was noticed that if 5 and T are regular, T acts on 5 and at least one of 5 and T is completely simple then Reg(5 * T) forms a (regular) subsemigroup in 5 * T, and the regular semidirect product S* r T of S by T was defined to be Reg(S* T).…”

“…Given any e-variety [pseudo-e-variety] U and a variety [pseudo-e-variety] V of completely simple semigroups, the regular semidirect product U * r V of U by V is defined in [9] to be the e-variety [pseudo-e-variety] generated by the class of all regular semidirect products T * r C with T eU and C e V.…”

“…Notice that if V £ Q and CZ £ V then the equality U * r V = U o r V does not hold in general. For example, Q * r UZ = CS, see [9], but Q o r UZ = Q v HZ. For, it is easy to check that, for any Pastijn-Rees product of a group G by a right zero semigroup A, we have G o r A = G o A, and the mapping G « A -> G x A, (g, k) i-> (gq\, k) is an isomorphism into the direct product of G and A.…”

“…Given a variety U of inverse semigroups and a variety V of completely simple semigroups with WB c V, the main result Theorem 3.9 of the previous section allows us to construct a model of any bifree object in the e-variety U * r V by making use of the Pastijn-Rees product construction. Similarly to the models of the bifree objects found in [10], these models have a graphical interpretation which makes them more transparent than the models presented in [9,Section 8].…”