Term Definable Classes of Boolean Functions and Frame Definability in Modal Logic

We establish a connection between term definability of Boolean functions and definability of finite modal frames. We introduce a bijective translation between functional terms and uniform degree-1 formulas and show that a class of Boolean functions is defined by functional terms if and only if the corresponding class of Scott-Montague frames is defined by the translations of these functional terms, and vice versa. As a special case, we get that the clone Λ1 of all conjunctions corresponds to the class of all Kripke frames. We also characterize some classes of Scott-Montague frames corresponding to subclones of Λ1 by restricting the class of Kripke frames in a natural way. Furthermore, by modifying Kripke semantics, we extend our results to correspondences between linear clones and classes of Kripke frames equipped with modified semantics.