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 , T Λ ) called a c -triple. In this study appear a number of relations on the free lattice generated by Γ. 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 of finite direct products, construct examples, and supplement some results in the paper referred to above.