Abstract
Given a class C of languages, let Pol( C ) be the polynomial closure of C , that is, the smallest class of languages containing C and closed under the operations union and marked product LaL', where a is a letter. We determine the polynomial closure of various classes of rational languages and we study the properties of polynomial closures. For instance, if C is closed under quotients (resp. quotients and inverse morphism) then Pol( C ) has the same property. Our main result shows that if C is a boolean algebra closed under quotients then Pol( C ) is closed under intersection. As an application, we refine the concatenation hierarchy introduced by Straubing and we show that the levels 1 2 and 3 2 of this hierarcy are decidable.
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