A generalization of the Schützenberger product of finite monoids

Abstract

For each n⩾1, an n-ary product ♢ on finite monoids is constructed. This product has the following property: Let Σ be a finite alphabet and Σ ∗ the free monoid generated by Σ. For i = 1, …, n, let A i be a recognizable subset of Σ ∗, M( A i ) the syntactic monoid of A n and M( A 1⋯ A n ) the syntactic monoid of the concatenation product A 1⋯ A n . Then M( A 1⋯ A n )< ♢ ( M( A 1),…, M( A n )). The case n = 2 was studied by Schützenberger. As an application of the generalized product, I prove the theorem of Brzozowski and Knast that the dot-depth hierarchy of star-free sets is infinite.