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.