Completely simple semigroups with nilpotent structure groups. (Q958198)

The author extends theorems of \textit{N. R. Reilly} and \textit{M. Petrich} [Trans. Am. Math. Soc. 280, 623-636 (1983; Zbl 0537.20031)] on varieties of central completely simple semigroups: those for which any product of idempotents lies in the centre of its containing subgroup. That situation clearly includes the completely simple semigroups all of whose subgroups are Abelian. Here an identity is found determining the completely simple semigroups whose subgroups are nilpotent of class \(\leq c\), for any \(c\); it is based on a semigroup identity that determines the corresponding variety of groups, found by \textit{B. H. Neumann} and \textit{T. Taylor} [Proc. R. Soc. Lond., Ser. A. 274, 1-4 (1963; Zbl 0115.02502)]. In addition, the notion of `central' is generalized to `\(w\)-marginal' in a related fashion and the theorem of Reilly and Petrich on central completely simple semigroups is extended similarly.