Special Congruence Triples for a Regular Semigroup

With the usual notation for congruences on a regular semigroup S, in a previous communication we studied the lattice Λ generated by Γ = {σ, τ, μ, β} relative to properties such as distributivity and similar conditions. For K and T the kernel and trace relations on the congruence lattice of S, we form an abstraction of the triple (Λ;K|Λ, TΛ) called a c-triple. In this study a number of relations on the free lattice generated by Γ appears. Here we study implications and independence of these relations, both on c-triples as well as on congruence lattices of regular semigroups. We consider the behavior of the members of Γ under forming finite direct products, construct examples and supplement some results in the paper referred to above.