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

Manfred Droste () and Paul Gastin ()
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Manfred Droste: Technische Universität Dresden, Institut für Algebra
Paul Gastin: Université Paris 7, LIAFA

A chapter in Formal Power Series and Algebraic Combinatorics, 2000, pp 158-169 from Springer

Abstract: 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.

Keywords: Formal Power Series; Free Monoid; Commutative Monoid; Characteristic Series; Trace Theory (search for similar items in EconPapers)
Date: 2000
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-662-04166-6_14

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DOI: 10.1007/978-3-662-04166-6_14

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