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Maximal groups in free Burnside semigroups

  • Alair Pereira do Lago
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1380)

Abstract

We prove that any maximal group in the free Burnside semigroup defined by the equation x n =x n+m for any n ≥ 1 and any m ≥ 1 is a free Burnside group satisfying x m =1. We show that such group is free over a well described set of generators whose cardinality is the cyclomatic number of a graph associated to the ℑ-class containing the group. For n=2 and for every m ≥ 2 we present examples with 2m−1 generators. Hence, in these cases, we have infinite maximal groups for large enough m. This allows us to prove important properties of Burnside semigroups for the case n=2, which was almost completely unknown until now. Surprisingly, the case n=2 presents simultaneously the complexities of the cases n=1 and n ≥ 3: the maximal groups are cyclic of order m for n ≥ 3 but they can have more generators and be infinite for n ≤ 2; there are exactly 2¦A¦ ℑ-classes and they are easily characterized for n=1 but there are infinitely many, ℑ-classes and they are difficult to characterize for n ≥ 2.

Keywords

Word Problem Local Group Short Word Congruence Class Finiteness Property 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 1998

Authors and Affiliations

  • Alair Pereira do Lago
    • 1
  1. 1.Instituto de Matemática e EstatísticaUniversidade de SÃo PauloSÃo Paulo - SPBrazil

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