Kernel normal systems for a semigroup.
The authors define a kernel normal system in a semigroup \(S\) to be the set of all idempotent congruence classes of some congruence on \(S\). It has long been known that on regular semigroups, each congruence is determined by its associated kernel normal system. Clearly, that need not be true in general, since for any congruence for which the quotient semigroup is idempotent-free, the kernel normal system will be empty. It is shown that every kernel normal system corresponds to an interval of congruences on \(S\). If \(\rho\) is a regular congruence on \(S\) (in the sense that \(S/\rho\) is a regular semigroup) then it is the greatest member of that interval. Hence regular congruences on \(S\) are determined by their kernel normal systems. The authors go on to consider the lattice of kernel normal systems and its relationship to the lattice of congruences.
scientific article; zbMATH DE number 5141237