All splitting logics in the lattice NEXT(KTB:30A)

We examine a special modal logic which is a normal extension of the Brouwer modal logic. It is determined by linearly ordered chains of clusters and the relation between clusters is reflexive and symmetric. The appropriate axiomatization of this logic is proposed in the papers [11] and [12]. There is also proved that all normal extensions of the investigated logic are Kripke complete and have f.m.p. Unfortunately, the cardinality of this family is continuum [13]. One may imagine that the structure of the lattice of these extensions is immensely complex. Then we use the technics of splitting to characterize this lattice and to describe some quite simple fragments. We characterize all the logics that split the lattice.