Decomposition of regular subsemigroup lattices
In an earlier paper [Algebra Univers. 30, 463-468 (1993; Zbl 0797.20046)], the author showed that for any \(E\)-solid or locally inverse regular semigroup \(S\), the set of regular subsemigroups forms a complete lattice \({\mathcal L}{\mathcal R}(S)\). The author considers the decomposition of this lattice induced by the relation \(\theta\) that identifies two regular subsemigroups if and only if they have the same idempotents. Those sets of idempotents that so arise are termed `closed' and have been characterized in various ways. They naturally form a complete lattice \({\mathcal L}{\mathcal C}(E)\). That \(\theta\) is a complete congruence follows from the fact that the map which assigns to each regular subsemigroup its set of idempotents is a complete morphism from \({\mathcal L}{\mathcal R}(S)\) onto \({\mathcal L}{\mathcal C}(E)\). Each \(\theta\)-class is isomorphic to the lattice of full regular subsemigroups of its maximum element and it is thus of interest to note that the latter is a complete sublattice of the lattice of all subsemigroups. The author points out a parallel with the `trace' congruence on the lattice of congruences of a regular semigroup.
scientific article; zbMATH DE number 611122
scientific article