  The wealth of information that is available on the lattice of varieties of
bands, is used to illuminate the structure of the lattice of
sub-pseudovarieties of DA, a natural generalization of bands which plays an
important role in language theory and in logic. The main result describes a
hierarchy of decidable sub-pseudovarieties of DA in terms of iterated Mal'cev
products with the pseudovarieties of definite and reverse definite semigroups.
