Bridges for Concatenation Hierarchies

In the seventies, several classification schemes for the rational languages were proposed, based on the alternate use of certain operators (union, complementation, product and star). Some thirty years later, although much progress has been done, several of the original problems are still open. Furthermore, their significance has grown considerably over the years, on account of the successive discoveries of surprising links with other fields, like non commutative algebra, finite model theory, structural complexity and topology. In this article, we solve positively a question raised in 1985 about concatenation hierarchies of rational languages, which are constructed by alternating boolean operations and concatenation products. We establish a simple algebraic connection between the Straubing-Thérien hierarchy, whose basis is the trivial variety, and the group hierarchy, whose basis is the variety of group languages. Thanks to a recent result of Almeida and Steinberg, this reduces the decidability problem for the group hierarchy to a property stronger than decidability for the Straubing-Thérien hierarchy. The reader is referred to [20] for undefined terms and a general overview of the motivations of this paper.