Hierarchies of Piecewise Testable Languages

The classes of languages which are boolean combinations of languages of the form Aa1A a2A ∗ . . . Aa`A , where a1, . . . , a` ∈ A, ` ≤ k , for a fixed k ≥ 0, form a natural hierarchy within piecewise testable languages and have been studied in papers by Simon, Blanchet-Sadri, Volkov and others. The main issues were the existence of finite bases of identities for the corresponding pseudovarieties of monoids and generating monoids for these pseudovarieties. Here we deal with similar questions concerning the finite unions and positive boolean combinations of the languages of the form above. In the first case the corresponding pseudovarieties are given by a single identity, in the second case there are finite bases for k equal to 1 and 2 and there is no finite basis for k ≥ 4 (the case k = 3 remains open). All the pseudovarieties are generated by a single algebraic structure.