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Almeida's generalised variety problem


Churchill, GA, Almeida's generalised variety problem, Journal of Algebra, 196, (2) pp. 499-519. ISSN 0021-8693 (1997) [Refereed Article]

DOI: doi:10.1006/jabr.1997.7037


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.

Item Details

Item Type:Refereed Article
Research Division:Mathematical Sciences
Research Group:Pure mathematics
Research Field:Algebra and number theory
Objective Division:Expanding Knowledge
Objective Group:Expanding knowledge
Objective Field:Expanding knowledge in the mathematical sciences
UTAS Author:Churchill, GA (Mrs Genevieve Annette Churchill)
ID Code:11508
Year Published:1997
Web of Science® Times Cited:2
Deposited By:Mathematics
Deposited On:1997-08-01
Last Modified:2011-08-12

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