Hiérarchies de Concaténation

As well-known, 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 Schützenberger product of the monoids M0, . . . , Mn, then L belongs to the Boolean closure of the set of languages of the form Li0a1Li1 · · · arLir (0 6 i0 < i1 < . . . < ir 6 n) where the ak are letters and the Lik are recognized by Mik (0 6 k 6 r). Decidability and inclusion problems are also discussed.