Polynomial Operators on Classes of Regular Languages

We assign to each positive variety V and a fixed natural number k the class of all (positive) boolean combinations of the restricted polynomials, i.e. the languages of the form L0a1L1a2 . . . a`L`, where ` ≤ k, a1, . . . , a` are letters and L0, . . . , L` are from the variety V. For this polynomial operator we give a certain algebraic counterpart which works with identities satisfied by syntactic (ordered) monoids of considered languages. We also characterize the property that a variety of languages is generated by a finite number of languages. We apply our constructions for particular examples of varieties of languages which are crucial for a certain famous open problem concerning concatenation hierarchies.