A special sublattice of the congruence lattice of a regular semigroup

Let S be a regular semigroup and C(S) be its congruence lattice. For rho is an element of C(S), we consider the sublattice L-rho of C(S) generated by the congruences rho w when w is an element of {K, k, T, t}* and w has no subword of the form KT, TK, kt, tk. Here K, k, T, t are the operators on C(S) induced by the kernel and the trace relations on C(S). We find explicitly the least lattice L whose homomorphic image is L-rho for all rho is an element of C(S) and represent it as a distributive lattice in terms of generators and relations. We also consider special cases: bands of groups, E-unitary regular semigroups, completely simple semigroups, rectangular groups as well as varieties of completely regular semigroups.