  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 and their intersections K\capTl,K\cap Tr . However, with the
exception of the lattice of all band varieties which happens to coincide with
the kernel class of the trivial variety, almost nothing is known about the
internal structure of individual K-classes beyond the fact that they are
intervals in L(CR). Here we present a number of general results that are
pertinent to the study of K -classes. This includes a variation of the renowned
Polak Theorem and its relationship to the complete retraction V -> V \cap B ,
where B denotes the variety of bands. These results are then applied, here and
in a sequel, to the detailed analysis of certain families of K -classes. The
paper concludes with results hinting at the complexity of K -classes in
general, such as that the classes of relation K/Kl may have the cardinality of
the continuum.
