The interpolation problem in finite-layered pre-Heyting logics

The interpolation problem over Johansson's minimal logic $\mathrm{ J}$ is considered. We introduce a series of Johansson algebras, which will be used to prove a number of necessary conditions for a $\mathrm{ J}$-logic to possess Craig's interpolation property $\mathrm{ (CIP)}$. As a consequence, we deduce that there exist only finitely many finite-layered pre-Heyting algebras with $\mathrm{ CIP}$.