Actions, wreath products of C-varieties and concatenation product

The framework of C-varieties, introduced by the third author, extends the scope of Eilenberg’s variety theory to new classes of languages. In this paper, we first define C-varieties of actions, which are closely related to automata, and prove their equivalence with the original definition of C-varieties of stamps. Next, we complete the study of the wreath product initiated by Ésik and Ito by extending its definition to C-varieties in two different ways, which are proved to be equivalent. We also state an extension of the wreath product principle, a standard tool of language theory. Finally, our main result generalizes to C-varieties the algebraic characterization of the closure under product of a variety of languages. Through the work of Eilenberg [3] and Schützenberger [13], the theory of varieties of finite semigroups and monoids emerged as an essential tool in the study of the algebra underlying families of regular languages. The current literature on the subject (see [10, 2] for a comprehensive bibliography) attests to the richness of this theory and the diversity of its applications in an increasing number of research fields including automata theory and formal languages but also model theory and logic, circuit complexity, communication complexity, discrete dynamical systems, etc. However, some important families of languages arising from open problems in language theory (the generalized star height problem), logic and circuit complexity [17], do not form varieties of languages in the sense originally described by Eilenberg. To study these new varieties of languages, Straubing [18] recently introduced the notion of C-varieties. A similar notion was introduced independently by Ésik and Ito [6]. The formal definition of a C-variety of languages is quite similar to Eilenberg’s except that it only requires closure under inverse images of morphisms belonging to some natural class C. (In the important applications, this class C is typically either the class of all length-preserving morphisms, or of all length-multiplying morphisms. In contrast, the theory developed by Eilenberg requires closure under inverse images of LIAFA, Université Paris VII and CNRS, Case 7014, 2 Place Jussieu, 75251 Paris Cedex 05, France. [email protected] LIAFA, Université Paris VII and CNRS, Case 7014, 2 Place Jussieu, 75251 Paris Cedex 05, France. [email protected] Department of Computer Science, Boston College, Chestnut Hill, MA 02167, USA [email protected]