Certain relations on the congruence lattice of a completely regular semigroup

Abstract

Let S be a completely regular semigroup, $\mathcal{C}\left( S \right)$ its congruence lattice and $\mathcal{P}$ be any of the Green relations $\mathcal{H},\mathcal{L},\mathcal{R}$ or $\mathcal{D}$ on S: The relation $\mathcal{P}^ \wedge$ defined on $\mathcal{C}\left( S \right)$ by: $\lambda \mathcal{P}^ \wedge \rho$ if $\lambda \cap \mathcal{P} = \rho \cap \mathcal{P}$ is a complete ^-congruence on $\mathcal{C}\left( S \right)$ but in general it is not a $\vee$-congruence. Semigroups S for which $\mathcal{P}^ \wedge$ is a congruence are characterized here in several ways. One of these characterizations consists of restricting the form of all congruences on S: In a general completely regular semigroup S; the set of all congruences of this type forms a complete sublattice of $\mathcal{C}\left( S \right)$ with some remarkable properties.