We study the relation B in le-semigroups that mimics the relation "to generate the same bi-ideal" considered by Kapp [1] in the plain semigroup case. This relation in general is finer than the Green-Kehayopulu relation H we studied in [6] and turns out to have better properties. In particular, the regularity (intra-regularity) of an element induces the regularity (respectively intra-regularity) of the whole B-class containing that element. We also provide various conditions under which B-classes are subsemigroups.