Abstract. A congruence $\rho $ on a regular semigroup $S$ is completely determined by its kernel and its trace. These two parameters induce the kernel relation $K$ and the trace relation $T$ on the lattice ${\cal C}(S)$ of congruences on $S$. The former is a complete $\wedge $-congruence and the latter is a complete congruence on ${\cal C}(S)$. The relation $K$ is generally not a $\vee $-congruence. We provide a number of necessary and sufficient conditions on $S$ for $K$ to be a congruence in terms of conditions on congruences, their kernels and their traces. When $K$ is a congruence, we consider conditions on $S$ which ensure that ${\cal C}(S)/K$ be modular. We conclude by examining the closure properties of the class of all regular semigroups $S$ for which $K$ is a congruence on ${\cal C}(S)$.

AMS Subject Classification (1991): 20M10, 20M17

Keyword(s): regular semigroup, congruence, kernel, trace, lattice congruence, modular, closure properties

Received April 1, 2003, and in final form April 14, 2004. (Registered under 5830/2009.)