Strictly positive fragments of K4 and its extensions

We consider two phenomena together. First, for any modal logic we can consider its strictly positive fragment, and sometimes two different modal logics have the same fragment; we then call them strictly positively equivalent. Second, all the normal extensions of K4 can be effectively classified by Zakharyashev's notation of canonical frame formulas. We would like then to characterize the set of normal modal logics strictly positively equivalent to K4. Obvilusly, K4 is the minimal logic in this set, so we are more interested in its maximal elements. In this talk we will present a partial answer on how this set looks like; in particular, we will prove that it does not have a unique greatest element.