THE GREEN RELATIONS APPROACH TO CONGRUENCES ON COMPLETELY REGULAR-SEMIGROUPS

On the congruence lattice C(S) of a completely regular semigroup S the following mappings are considered chi(p):rho --> rho boolean AND P and chi(P):rho boolean OR P, where P is any of the Green relations H, L, R or D. The equivalence relations P-boolean AND and P-boolean OR induced by these maps represent the main object of study in the paper. The former is a complete boolean AND-congruence whereas the latter is a complete congruence on C(S). In particular H-boolean AND, H-boolean OR, L(boolean OR), R(boolean OR) coincide with the kernel, trace, left trace and right trace relations on C(S), respectively. All essential properties known for the latter relations carry over to the new relations P-boolean AND and P-boolean OR. In addition, some interesting interplays of these provide for more richness in the theory of congruences on completely regular than is the case for the kernel-trace approach to congruences on regular semigroups.