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On Aperiodic and Star-free Formal Power Series in Partially Commuting Variables

In: Formal Power Series and Algebraic Combinatorics

Author

Listed:
  • Manfred Droste

    (Technische Universität Dresden, Institut für Algebra)

  • Paul Gastin

    (Université Paris 7, LIAFA)

Abstract

Formal power series over non-commuting variables have been investigated as representations of the behavior of automata with multiplicities. Here we introduce and investigate the concepts of aperiodic and of star-free formal power series over semirings and partially commuting variables. We prove that if the semiring K is idempotent and commutative, or if K is idempotent and the variables are non-commuting, then the product of any two aperiodic series is again aperiodic. We also show that if K is idempotent and the matrix monoids over K have a Burnside property (satisfied, e.g. by the tropical semiring), then the aperiodic and the star-free series coincide. This generalizes a classical result of Schützenberger (1961) for aperiodic regular languages and contains a result of Guaiana, Restivo and Salemi (1992) on aperiodic trace languages.

Suggested Citation

  • Manfred Droste & Paul Gastin, 2000. "On Aperiodic and Star-free Formal Power Series in Partially Commuting Variables," Springer Books, in: Daniel Krob & Alexander A. Mikhalev & Alexander V. Mikhalev (ed.), Formal Power Series and Algebraic Combinatorics, pages 158-169, Springer.
    • Handle: RePEc:spr:sprchp:978-3-662-04166-6_14
    DOI: 10.1007/978-3-662-04166-6_14
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