The concept of a completely regular semigroup was first introduced by Lyapin (Semigroups, Fizmatgiz, Moscow (1960) (in Russian); English transl. (2nd edn.), American Mathematical Society, Providence, RI (1968)). Since then, a large literature has developed devoted to the investigation of the structure of and congruences on completely regular semigroups as well the lattice \(\mathcal {L(CR)}\) of varieties of completely regular semigroups. The theorem of Polák that provides a subdirect product representation of the sublattice of varieties containing the variety of semilattices could be considered as the theory’s crown jewel. Complicated as this theorem may appear, a vast amount of information about special sublattices can be gleaned from it. Polák’s Theorem depends on a theory of special operators on the lattice \(\mathcal {L(CR)}\) and the varieties that are invariant under these operators constitute a bit of a blind spot, so to speak, in the application of Polák’s Theorem. Recent investigations into the structure of kernel classes of varieties of completely regular semigroups have taken advantage of the Polák representation of the lattice of varieties of completely regular semigroups. This approach uses the restrictions of the left and right trace relations to individual kernel classes, namely the left and right kernel relations together with their associated upper and lower operators. However, in order to be successful in completely describing a kernel class, it is helpful to be able to reach each variety in the kernel class by a succession of applications of associated left and right kernel operators and their intersections. Certain descending chains of varieties are essential to Polák’s Theorem. We consider the question of their finiteness and the involvement of non-finitely based varieties.
