Canonical Varieties of Completely Regular Semigroups

Completely regular semigroups CR are regarded here as algebras with multiplication and the unary operation of inversion. Their lattice of varieties is denoted by L.CR/. Let B denote the variety of bands and L.B/ the lattice of its subvarieties. The mapping V → V ∩ B is a complete homomorphism of L.CR/ onto L.B/. The congruence induced by it has classes that are intervals, say VB = [VB ;V B] for V ∈ L.CR/. Here VB = V ∩ B. We characterize V B in several ways, the principal one being an inductive way of constructing bases for ∨-irreducible band varieties. We term the latter canonical. We perform a similar analysis for the intersection of these varieties with the varieties BG, OBG and B. 2000 Mathematics subject classification: primary 20M07.