Rees Matrix Covers and Semidirect Products of Regular Semigroups

✍ Scribed by Peter R. Jones

• Publisher: Elsevier Science
• Year: 1999
• Tongue: English
• Weight: 138 KB
• Volume: 218
• Category: Article
• ISSN: 0021-8693
• DOI: 10.1006/jabr.1999.7872

✦ Synopsis

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 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.