New operators for varieties of completely regular semigroups

Operators K, T, L and C on the lattice L(CR) of varieties of completely regular semigroups have been successfully used in the study of global and local structure of L(CR). They are either induced by relations with origin in the theory of congruences on regular semigroups or are directly defined on L(CR). We introduce new relations B-Lambda, B-Lambda, G(Lambda), G(Lambda), CS Lambda, CS Lambda, induced by varieties B, G, CS, and to each associate lower and upper operators. For example, for U, V is an element of L(CR), U B-Lambda V if U boolean AND B = V boolean AND B, and the operators b(Lambda) : V -> V boolean AND B and B-Lambda : V -> W where W is the greatest member of the B.-class containing V. We set up lattice theoretical foundation for these relations and operators and study their relationship both mutual and with the well known operators mentioned above on various parts of L(CR), in particular on sublatticesL(O), L(BG), L(OBG) andL(LOBG). Composition of these operators, new and old, produce formulae which are very useful in their manipulation, for instance, commutativity of certain pairs. Four diagrams serve to illustrate some of the results of the paper.