Congruence lattices of regular semigroups related to kernels and traces

Abstract

The kernel-trace approach to congruences on a regular semigroup S can be refined by introducing the left and right traces. This induces eight operators on the lattice of congruences on S: t,, k, t r ; T h K, T r ; t, T. We describe the lattice of congruences on S generated by six 3-element subsets of the set {cot,, cok, a>t r , eT,, e.K, eT,} where co and E denote the universal and the equality relations. This is effected by means of a diagram and in terms of generators and relations on a free distributive lattice, or a homomorphic image thereof. We perform the same analysis for the lattice of congruences on S generated by the set {eK, cok, ET, cot}.