Given a normal (multi-)modal logic a characterization is given of the finitely presentable algebras A whose logics L A split the lattice of normal extensions of . This is a substantial generalization of Rautenberg [10] and [11] in which is assumed to be weakly transitive and A to be finite. We also obtain as a direct consequence a result by Blok [2] that for all cycle-free and finite A L A splits the lattice of normal extensions of K. Although we (...) firmly believe it to be true, we have not been able to prove that if a logic splits the lattice of extensions of then is the logic of an algebra finitely presentable over ; in this respect our result remains partial. (shrink)

For a novice this book is a mathematically-oriented introduction to modal logic, the discipline within mathematical logic studying mathematical models of reasoning which involve various kinds of modal operators. It starts with very fundamental concepts and gradually proceeds to the front line of current research, introducing in full details the modern semantic and algebraic apparatus and covering practically all classical results in the field. It contains both numerous exercises and open problems, and presupposes only minimal knowledge in mathematics. A specialist (...) can use the book as a source of references. Results and methods of many directions in propositional modal logic, from completeness and duality to algorithmic problems, are collected and systematically presented in one volume. (shrink)