Kernel classes of varieties of completely regular semigroups II

Three complete congruences on the lattice L(CR) of varieties of completely regular semigroups have been fundamental to studies of the structure of L(CR). These are the kernel relation K, the left trace relation Tl and the right trace relation Tr. However, with the exception of the lattice of all band varieties L(B), which happens to coincide with the kernel class of the trivial variety, very little has been written about the structure of individual K-classes beyond the fact that they are intervals in L(CR). Here we build on the general background developed in Reilly (Semigroup Forum, 2019, https://doi.org/10.1007/s00233-019-10056-7) to study the structure of certain specific K-classes. We provide an exact description of those kernel classes that contain a variety of abelian groups of exponent a prime. In doing so we provide exact descriptions of various intervals in L(CR) which lie entirely above the variety of all bands.