Rees matrix covers and semidirect products of regular semigroups
In this important paper a regular version of Tilson's Delay Theorem is presented [see \textit{B. Tilson}, J. Pure Appl. Algebra 48, 83-198 (1987; Zbl 0627.20031)]. Let \(S\) and \(T\) be regular semigroups, one of them being completely simple and \(T\) acting on \(S\) by endomorphisms on the left. Then within the usual semidirect product \(S*T\) lies the set of all regular elements \(\text{Reg}(S*T)\) which forms a (necessarily regular) subsemigroup. This leads to a definition of a semidirect product \(\mathbf{U*V}\) of e-varieties \(\mathbf U\) and \(\mathbf V\) in a natural way (with the only restriction imposed being that \(*\) is defined only in case that either \(\mathbf U\) or \(\mathbf V\) entirely consists of completely simple semigroups) [for a development of this theory see \textit{P.~R.~Jones} and \textit{P.~G.~Trotter}, Trans. Am. Math. Soc. 349, No. 11, 4265-4310 (1997; Zbl 0892.20037)]. In the present paper those e-varieties \(\mathbf V\) are characterized for which the equation \(L\mathbf{V=V*RZ}\) holds where \(L\) is the usual ``local'' operator and \(\mathbf{RZ}\) denotes the e-variety of all right zero semigroups. The equation is true precisely if every regular category with local monoids in \(\mathbf V\) regularly divides a member of \(\mathbf V\); in turn, this is shown to be equivalent that each regular semigroup with all local submonoids in \(\mathbf V\) regularly divides a regular Rees matrix semigroup over a member of \(\mathbf V\). Reviewer's remark: There is a slight inaccuracy in the proofs of Proposition 1.9 and Theorem 2.1 which is in turn based on an erroneous observation on p.~291 stating that the e-variety generated by a class \(\mathcal C\) of regular semigroups consists of all regular semigroups that are obtained by a finite chain of divisions of direct products of members of \(\mathcal C\). However, there is no plausibility nor an argument that the class so obtained is closed under direct products and so is an e-variety at all. In the forthcoming paper ``Associativity of the regular semidirect product of existence varieties'' by \textit{B.~Billhardt} and \textit{M.~B.~Szendrei} [J. Aust. Math. Soc. (to appear)] a transfinite procedure is discussed which produces the least e-variety \(\langle\mathcal C\rangle\) containing a given class \(\mathcal C\). Fortunately, Proposition 1.9 and Theorem 2.1 nevertheless are true. On the one hand, this could be seen by incorporating the appropriate transfinite arguments in the respective proofs. On the other hand, the proofs themselves show that under the assumptions of the statements each finite chain of regular divisions may be replaced with just one such division, so that, in those particular cases a usual \(HSP\)-theorem holds.
scientific article; zbMATH DE number 1348161