Even more about the lattice of tense logics

The present paper is based on [11], where a number of conjectures are made concerning the structure of the lattice Ext(Kt) of normal extensions of the tense logic Kt. That paper was mainly dealing with splittings of Ext(Kt) and some sublattices, and this is what I will concentrate on here as well. The main tool in analysing the splittings of Ext(Kt) will be the splitting theorem of [7]. In [11] it was conjectured that each finite subdirectly irreducible algebra splits the lattice of normal extensions of K4t and S4t. We will show that this is not the case and that on the contrary only very few and trivial splittings of the mentioned lattices exist. ∗I wish to thank Prof. Rautenberg for suggesting this work to me and for waiting patiently for two years until I started it. Thanks also to two anonymous referees and Frank Wolter for helpful discussion of this paper. One of the referees deserves special mentioning for his precise and detailed criticism.