An almost general splitting theorem for modal logic

Given a normal (multi-)modal logic Θ, a characterization is given of the finitely presentable algebras A whose logics LA split the lattice of normal extensions of Θ. This is a substantial generalization of [Rautenberg, 1980; Rautenberg, 1977] in which Θ is assumed to be weakly transitive and A to be finite. We also obtain as a direct consequence a result by [Blok, 1978] that for all cycle-free and finite A LA 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 Λ splits the lattice of extensions of Θ then Λ is the logic of an algebra finitely presentable over Θ; in this respect our result remains partial.