Almeida's generalised variety problem

Let Com denote the variety of all commutative semigroups, let Nil denote the generalized variety of all nil semigroups, and let N denote the generalized variety of all nilpotent semigroups. For any class W of semigroups let L(W) denote the lattice of all varieties contained in W, and let G(W) denote the lattice of all generalized varieties contained in W. Almeida has shown that the map φ: L(Nil ∩ Com) ∪ {Nil ∩ Com} → G(N∩ Com) given by Wφ = W∩ N is an isomorphism, and asked whether the extension of this map to L(Nil) ∪ {Nil} is also an isomorphism. In this article a negative answer is given: two varieties U, V ∈ L(Nil) are defined and used to show that the map is not injective. In the process, congruence classes of the fully invariant congruences on the free semigroup on any countable set X with |X| ≥ 3 which correspond to U and V are described which are denumerable and contain words of unbounded length. © 1997 Academic Press.