THE KERNEL RELATION FOR A COMPLETELY REGULAR SEMIGROUP

The kernel relation for a regular semigroup S identifies two congruenc es on S if they have the same kernel. It is always a complete boolean AND-congruence on the congruence lattice C(S) of S. We give a great nu mber of equivalent conditions on a completely regular semigroup S, one of which is that K be a (complete) congruence on C(S). These conditio ns bear upon minimal congruences identifying two comparable elements o f S, variants of theta-modularity, the mappings rho --> ker rho, rho - -> rho(K), rho --> rho boolean AND H being (complete) boolean OR-homom orphisms, least group congruences on certain completely simple semigro ups, certain subgroups of S, and the standard representation of S. The paper concludes with a discussion of special cases. (C) 1995 Academic Press, Inc.