The lattice of pseudovarieties of idempotent semigroups and a non-regular analogue

We use classical results on the lattice ℒ (ℬ) of varieties of band (idempotent) semigroups to obtain information on the structure of the lattice Ps(DA) of subpseudovarieties of DA, - where DA is the largest pseudovariety of finite semigroups in which all regular semigroups are band semigroups. We bring forward a lattice congruence on Ps(DA), whose quotient is isomorphic to ℒ(ℬ), and whose classes are intervals with effectively computable least and greatest members. Also we characterize the pro-identities satisfied by the members of an important family of subpseudovarieties of DA. Finally, letting Vk be the pseudovariety generated by the k -generated elements of DA (k ≥ 1), we use all our results to compute the position of the congruence class of Vk in ℒ(ℬ).