Abstract :
Let imageimageimage denote the variety of all commutative semigroups, let imageimageℓ denote the generalized variety of all nil semigroups, and let image denote the generalized variety of all nilpotent semigroups. For any class image of semigroups let image(image) denote the lattice of all varieties contained in image, and let image(image) denote the lattice of all generalized varieties contained in image. Almeida has shown that the map phi: image(imageimageℓ ∩ imageimageimage) union or logical sum {imageimageℓ ∩ imageimageimage} → image(image ∩ imageimageimage) given by imagephi = image ∩ image is an isomorphism, and asked whether the extension of this map to image(imageimageℓ) union or logical sum {imageimageℓ} is also an isomorphism. In this article a negative answer is given: two varieties image, image set membership, variant image(imageimageℓ) 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 setXwith X ≥ 3 which correspond to image and image are described which are denumerable and contain words of unbounded length.
