Centrality in Rees-Sushkevich varieties

Abstract

Denote by RSn the variety generated by all completely 0-simple semigroups over groups of exponent dividing n. Subvarieties of RSn 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 CmRSn to be the class of all semigroups S in RSn 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 CmRSn constitute varieties that are the main object of investigation in this article. It is shown that a sublattice of exact subvarieties of C 2 RSn 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 CmRSn is also addressed. For any fixed n ≥ 2, it is shown that although the varieties CmRSn, where m = 1, 2, . . . , are all finitely based, their complete intersection (denoted by C∞RSn) is non-finitely based. Further, the variety C∞RSn contains a continuum of ultimately incomparable infinite sequences of finitely generated exact subvarieties that are alternately finitely based and non-finitely based.