${\mathcal C}$ -Varieties, Actions and Wreath Product

Abstract

Motivated by open problems in language theory, logic and circuit complexity, Straubing generalized Eilenberg's variety theory, introducing the ${\mathcal C}$-varieties. As a further contribution to this theory, this paper first studies a new ${\mathcal C}$-variety of languages, lying somewhere between star-free and regular languages. Then, continuing the early works of Esik-Ito, we extend the wreath product to ${\mathcal C}$-varieties and generalize the wreath product principle, a powerful tool originally designed by Straubing for varieties. We use it to derive a characterization of the operations L→ LaA* and L → La on languages. Finally, we investigate the decidability of the operation V →V∗LI (the wreath product by locally trivial semigroups) and solve it explicitely in several non-trivial cases.