On regularity preservation in a semigroup

We consider certain subsets of a semigroupS, defined mainly by conditions involving regularity preservation. In particular, theregular baseB(S) ofSmay be regarded as a generalisation of the zero ideal in a semigroup with zero; if it non-empty thenSisE-inversive. The other subsets considered are related in a natural way either to B(S) or to the set RP(S) of regularity-preserving elements inS. In a regular semigroup (equipped with the Hartwig-Nambooripad order) each of these subsets contains either minimal elements only or maximal elements only. The relationships between the subsets are discussed, and some characterisations of completely simple semigroups are obtained.