  The kernel relation $K$ on the lattice $\mathcal{L}(\mathcal{CR})$ of
varieties of completely regular semigroups has been a central component in many
investigations into the structure of $\mathcal{L}(\mathcal{CR})$. However,
apart from the $K$-class of the trivial variety, which is just the lattice of
varieties of bands, the detailed structure of kernel classes has remained a
mystery until recently. Kad'ourek [RK2] has shown that for two large classes of
subvarieties of $\mathcal{CR}$ their kernel classes are singletons. Elsewhere
(see [RK1], [RK2], [RK3]) we have provided a detailed analysis of the kernel
classes of varieties of abelian groups. Here we study more general kernel
classes. We begin with a careful development of the concept of duality in the
lattice of varieties of completely regular semigroups and then show that the
kernel classes of many varieties, including many self-dual varieties, of
completely regular semigroups contain multiple copies of the lattice of
varieties of bands as sublattices.
