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Maximal groups in free Burnside semigroups

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Book cover LATIN'98: Theoretical Informatics (LATIN 1998)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 1380))

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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.

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Authors and Affiliations

  1. Instituto de Matemática e Estatística, Universidade de SÃo Paulo, Rua do MatÃo, 1010, 05508-900, SÃo Paulo - SP, Brazil

    Alair Pereira do Lago

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  1. Alair Pereira do Lago
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Cláudio L. Lucchesi Arnaldo V. Moura

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© 1998 Springer-Verlag Berlin Heidelberg

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do Lago, A.P. (1998). Maximal groups in free Burnside semigroups. In: Lucchesi, C.L., Moura, A.V. (eds) LATIN'98: Theoretical Informatics. LATIN 1998. Lecture Notes in Computer Science, vol 1380. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0054311

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  • DOI: https://doi.org/10.1007/BFb0054311

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-64275-6

  • Online ISBN: 978-3-540-69715-2

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