Malcev products of unipotent monoids and varieties of bands

Abstract

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 $\mathcal{G}$.