Abstract
Completely regular semigroups are semigroups which are (disjoint) unions of groups. In this article we review the progress made in the study of the lattice L(CR) of varieties of completely regular semigroups. Various results describing the lattice of sub-varieties L(U) of some proper subvariety U of CR are given; for example, with U equal to the variety of bands, central completely simple semigroups and several other interesting varieties. In the final section, certain global decompositions of L(CR) using the concepts of kernel, left trace and right trace are discussed.
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