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  • 1 University of York Department of Mathematics Heslington, York YO10 5DD UK
  • 2 Centro de Álgebra da Universidade de Lisboa Avenida Prof Gama Pinto, 2 1649-003 Lisboa Portugal
  • 3 Universidade de Lisboa Departamento de Matemática, Faculdade de Ciências 1746-016 Lisboa Portugal
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Abstract  

We consider the question of membership of AG, where A and G are the pseudovarieties of finite aperiodic semigroups, and finite groups, respectively. We find a straightforward criterion for a semigroup S lying in a class of finite semigroups that are weakly abundant, to be in AG. The class of weakly abundant semigroups contains the class of regular semigroups, but is much more extensive; we remark that any finite monoid with semilattice of idempotents is weakly abundant. To study such semigroups we develop a number of techniques that may be of interest in their own right.