Hiérarchies de concaténation

Abstract

Is it well-known that there exists a one-to-one correspondence between varieties of recognizable languages and varieties of finite semigroups. New hierarchies of varieties of languages (based on the concatenation product) are defined and an algebraic description of the corresponding hierarchies of varieties of semigroups is given. Various well-known hierarchies are obtained as particular cases. The construction is based on the following result: if a language L is recognized by the Schutzenberger product of the monoids M_0, ..., M_n, then L belongs to the Boolean closure of the set of languages of the form L_{i_0}a_1L_{i_1} ... a_rL_{i_r} (0 ≤ i_0 < i_1 < ... < i_r ≤ n) where the ak are letters and the Lik are recognized by M_{i_k} (0 ≤ k ≤ r). Decidability and inclusion problems are also discussed.