Quasi-normal partners of modal logics K4 and GL

The paper considers properties of relational models for quasi-normal modal logics containing the transitivity formula $\Box p\to \Box\Box p$ and (or) the Löb formula $\Box(\Box p\to p)\to \Box p$. It is proved that the accessibility relation in refined relational models for quasi-normal companions of such logics as $\bf K4$ and $\bf GL$, as in the normal case, is transitive. Questions concerned axiomatization of the quasi-normal companion of $\bf GL$ under such logics as $\bf K4$ and $\bf K$ are considered. The following fragments are investigated: the fragment of the lattice of quasi-normal logics containing the transitivity formula and (or) the Löb formula and the fragment of the lattice of normal companions of these logics. We consider the function which maps a quasi-normal logic to its normal companion. It is proved that this function is a pseudo-epimorphism.