Free shuffle algebras in language varieties

Abstract

We give simple concrete descriptions of the free algebras in the varieties generated by the “shuffle semirings” L Σ := (P(Σ ∗),+,., ⊗, 0,1) , or the semirings R Σ := (R(Σ ∗),+,., ⊗, ∗,0,1) , where P(Σ ∗) is the collection of all subsets of the free monoid Σ ∗, and R(Σ ∗) is the collection of all regular subsets. The operation x ⊗ y is the shuffle product.