Abstract
Completely regular semigroups \mathcal {CR} are regarded here as algebras with multiplication and the unary operation of inversion. Their lattice of varieties is denoted by \mathcal L (\mathcal {CR}). Let \mathcal B denote the variety of bands and \mathcal L (\mathcal B) the lattice of its subvarieties. The mapping \mathcal V \rightarrow \mathcal V \cap \mathcal B is a complete homomorphism of \mathcal L (\mathcal {CR}) onto \mathcal L(\mathcal B). The congruence induced by it has classes that are intervals, say \mathcal {V}B=[\mathcal V_B,\mathcal V^B] for \mathcal V \in \mathcal L(\mathcal {CR}). Here \mathcal V_B=\mathcal V \cap \mathcal B. We characterize \mathcal V^B in several ways, the principal one being an inductive way of constructing bases for \vee irreducible band varieties. We term the latter canonical. We perform a similar analysis for the intersection of these varieties with the varieties \mathcal {BG}, \mathcal {OBG} and \mathcal B.
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