On Boolean Closed Full Trios and Rational Kripke Frames

We study what languages can be constructed from a non-regular language L using Boolean operations and synchronous or non-synchronous rational transductions. If all rational transductions are allowed, one can construct the whole arithmetical hierarchy relative to L . In the case of synchronous rational transductions, we present non-regular languages that allow constructing languages arbitrarily high in the arithmetical hierarchy and we present non-regular languages that allow constructing only recursive languages. A consequence of the results is that aside from the regular languages, no full trio generated by a single language is closed under complementation. Another consequence is that there is a fixed rational Kripke frame such that assigning an arbitrary non-regular language to some variable allows the definition of any language from the arithmetical hierarchy in the corresponding Kripke structure using multimodal logic.