The Kernel Relation for a Completely Regular Semigroup Original Research Article

The kernel relation for a regular semigroup S identifies two congruences on S if they have the same kernel. It is always a complete logical and-congruence on the congruence lattice image(S) of S. We give a great number of equivalent conditions on a completely regular semigroup S, one of which is that K be a (complete) congruence on image(S). These conditions bear upon minimal congruences identifying two comparable elements of S, variants of θ-modularity, the mappings ρ → ker ρ, ρ → ρK, ρ → ρ ∩ image being (complete) logical or-homomorphisms, least group congruences on certain completely simple semigroups, certain subgroups of S, and the standard representation of S. The paper concludes with a discussion of special cases.