Malcev products of unipotent monoids and varieties of bands

Let (mathcal{U}) be the class of all unipotent monoids and (mathcal{B}) the variety of all bands. We characterize the Malcev product (mathcal{U} circ mathcal{V}) where (mathcal{V}) is a subvariety of (mathcal{B}) low in its lattice of subvarieties, (mathcal{B}) itself and the subquasivariety (mathcal{S} circ mathcal{RB}), where (mathcal{S}) stands for semilattices and (mathcal{RB}) for rectangular bands, in several ways including by a set of axioms. For members of some of them we describe the structure as well. This succeeds by using the relation (widetilde{mathcal{H}}= widetilde{mathcal{L}} cap widetilde{mathcal{R}}), where (a;,widetilde{mathcal{L}};,b) if and only if a and b have the same idempotent right identities, and (widetilde{mathcal{R}}) is its dual.We also consider ((mathcal{U} circ mathcal{RB}) circ mathcal{S}) which provides the motivation for this study since ((mathcal{G} circ mathcal{RB}) circ mathcal{S}) coincides with completely regular semigroups, where (mathcal{G}) is the variety of all groups. All this amounts to a generalization of the latter: (mathcal{U}) instead of (mathcal{G}).