Centrality in Rees–Sushkevich varieties

Denote by RS_n the variety generated by all completely 0-simple semigroups over groups of exponent dividing n. Subvarieties of RS_n are called Rees–Sushkevich varieties and those that are generated by completely simple or completely 0-simple semigroups are said to be exact. For each positive integer m, define C_mRS_n to be the class of all semigroups S in RS_n with the property that if the product of m idempotents of S belongs to some subgroup of S, then the product belongs to the center of that subgroup.

The classes C_mRS_n constitute varieties that are the main object of investigation in this article. It is shown that a sublattice of exact subvarieties of C₂RS_n is isomorphic to the direct product of a three-element chain with the lattice of central completely simple semigroup varieties over groups of exponent dividing n. In the main result, this isomorphism is extended to include those exact varieties for which the intersection of the core with any subgroup, if nonempty, is contained in the center of that subgroup.

The equational property of the varieties C_mRS_n is also addressed. For any fixed n ≥ 2, it is shown that although the varieties C_mRS_n, where m = 1, 2,…, are all finitely based, their complete intersection (denoted by C_∞RS_n) is non-finitely based. Further, the variety C_∞RS_n contains a continuum of ultimately incomparable infinite sequences of finitely generated exact subvarieties that are alternately finitely based and non-finitely based.