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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. Kadourek [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.
Several complete congruences on the lattice L(CR) of varieties of completely regular semi- groups have been fundamental to studies of the structure of L(CR). These are the kernel relation K , the left trace relation Tl , the right trace relation Tr a
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