We introduce the syntax and the semantics of intuitionistic modal logics based on a diamond connective à la Prenosil, its dual box connective, a diamond connective à la Wijesekera and its dual box connective. We analyze the modal definability of some elementary classes of frames. We study the complete axiomatizability of the sets of valid formulas determined by these classes of frames. We prove the decidability of the minimal intuitionistic modal logic determined by the class of all frames.

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