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

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 Sch

• Type: Journal
• Year: 2008
• Where: MST
• Authors: Manfred Droste, Paul Gastin
• Keywords: Aperiodic, Aperiodic Series, Formal Power Series, MST 2008