The Green relations approach to congruences on completely regular semigroups (Q1342843)

Let \(S\) be a completely regular semigroup and \({\mathcal C}(S)\) its congruence lattice. Let \(\mathcal P\) be any of Green's relations \(\mathcal H\), \(\mathcal R\), \(\mathcal L\) or \(\mathcal D\) on \(S\). For \(\rho\), \(\lambda \in {\mathcal C}(S)\), put \(\rho {\mathcal P}^ \wedge \lambda\) [resp. \(\rho {\mathcal P}^ \vee \lambda]\) if and only if \(\rho \cap {\mathcal P} = \lambda \cap {\mathcal P}\) [resp. \(\rho \vee {\mathcal P} = \lambda \vee {\mathcal P}]\) in the lattice of equivalence relations. Then \({\mathcal P}^ \wedge\) is a complete \(\cap\)- congruence and \({\mathcal P}^ \vee\) is a complete congruence on \({\mathcal C}(S)\). For \(\rho \in {\mathcal C}(S)\) the \({\mathcal P}^ \wedge\)- and \({\mathcal P}^ \vee\)-class of \(\rho\) are intervals with smallest elements \(\rho_ p\) and \(\rho_ P\) respectively. The ends of such intervals are characterized. Several characterizations are given for the congruence relations which are \({\mathcal P}^ \wedge\)- or \({\mathcal P}^ \vee\)-related to the equality relation or the universal relation on \(S\) and for the congruence relations which are \({\mathcal P}^ \vee\)-related to the congruence relation \({\mathcal P}^*\) which is generated by \(\mathcal P\). For \(\rho \in C(S)\), the sublattice of \({\mathcal C}(S)\) generated by \(\rho, \rho_ p, \rho_ P, \rho_{pP}, \rho_{Pp}, \rho_{pPp}, \dots,\) is investigated.