Certain relations on the congruence lattice of a completely regular semigroup (Q1300602)

Let \(S\) be a completely regular semigroup and let \(\mathcal P\) stand for any one of Green's relations \(\mathcal{H,L}\) and \(\mathcal D\) on \(S\) (noting that \({\mathcal J}={\mathcal D}\) and \(\mathcal R\) will be treated dually). The relation \({\mathcal P}^\wedge\) that is defined on the congruence lattice \({\mathcal C}(S)\) of \(S\) by \(\lambda{\mathcal P}^\wedge\rho\) if \(\lambda\cap{\mathcal P}=\rho\cap{\mathcal P}\), is a meet-congruence but not, in general, a congruence. Necessary and sufficient conditions are given in order that this relation should indeed be a congruence. An example of such a condition is that if \(a,b\in S\) and \(a>b\) in the natural partial order, then the \(\mathcal P\)-class of \(a\) should be contained in its \(\rho\)-class, whenever \(a\) and \(b\) are related under some congruence \(\rho\). In the special case when \(S\) is a normal cryptogroup, that is, a normal band of groups, then these conditions may be stated in terms of the natural structure mappings: for each \(\mathcal D\)-class \(D\) of \(S\), the restriction of \(\mathcal P\) to \(D\) must be contained in the kernel of every structure mapping whose domain is \(D\). Various other situations are also considered.