The kernel relation for a completely regular semigroup
The kernel of a congruence \(\rho\) on a regular semigroup \(S\) is the union of the idempotent \(\rho\)-classes. The kernel relation \(K\) is the equivalence relation on the congruence lattice \(C(S)\) of \(S\) that identifies those congruences with the same kernel. This relation was introduced by \textit{F. Pastijn} and the author [Trans. Am. Math. Soc. 295, 607-633 (1986; Zbl 0599.20095)], who showed that it is \(\wedge\)- preserving but not, in general, \(\vee\)-preserving. The author gives a large number of conditions that are equivalent to the latter property (that is, that \(K\) be a congruence on \(C(S)\)), for completely regular semigroups. Some of these are related to conditions given earlier by various authors in connection with modularity of \(C(S)\). If \(K\) is indeed a congruence then it is necessarily complete and \(C(S)/K\) turns out to be a modular lattice; various other consequences are found. The condition that \(K\) be a congruence is also expressed in terms of the author's `standard' model of completely regular semigroups.
scientific article; zbMATH DE number 763908