Products of group languages

Concatenation product is, together with Kleene’s star operation, one of the most fascinating operations on recognizable (= rational, regular) languages. The study of this operation produced numerous fundamental results like Schützenberger’s theorem on star-free languages, Brzozowski’s results on the dot depth hierarchy, Simon’s theorem on piecewiese testable languages, Straubing’s characterization of varieties closed under product, results of Mc Naughton and Thomas on the connexion with first order logic, etc. It also had a considerable influence on the rest of the theory and many algebraic tools were originally introduced to produce better proofs of old or new results. In fact, it appears that the trully fundamental operation is not exactly the usual concatenation product but a variant of it, that consists to associate to languages L0, L1, . . . , Ln the language L0a1L1a2 · · · anLn where a1, a2, . . . , an are given letters of the alphabet. Notice that this operation is not mysterious at all. It is used for instance to obtain a rational expression associated to a finite automaton in the classical algorithm of Mc Naughton and Yamada. Therefore, in this paper, the term “product” will refer to this variant of concatenation product. With this operation in hand, it is not difficult to construct hierarchies of recognizable languages. Start with a boolean algebra of languages: this will be the level 0 of our hierarchy. Then define level n+ 1 as the boolean algebra generated by products (in the new sense) of languages of level n. If you start with the trivial boolean algebra {∅, A}, you obtain Straubing’s hierarchy. If you start with endwise testable (or “generalized definite”) languages, you get Brzozowski’s hierarchy, also called dot-depth hierarchy. The aim of this paper is to study the hierarchy whose level 0 consists of all group languages. In this case, the union U of all levels of the hierarchy is the closure of group languages under product and boolean operations. Our first result shows that U is a decidable variety of languages. That is, given a recognizable language L, one can decide whether L belongs to U or not. Our second result states that our hierarchy is strict. In fact, this result still holds if one takes as level 0 an arbitrary subvariety of the variety of group languages. The rest of the paper is devoted to the study of level 1. It turns out that this variety of languages, and the corresponding variety of monoids ♦G, appear in many different contexts. First, ♦G is exactly the variety J ∗G generated by all semidirect products of a J -trivial monoid by a group. This result is interesting because J is also the first level of Straubing’s hierarchy Vn. Thus, at least for